diagnosability of star graphs with missing edges in this paper we study the system diagnosis on an n dimensional star under the comparison model following the concept of local diagnosability 3 the strong local diagnosability property 7 is discussed this property describes the equivalence of the local diagnosability of a node and its degree we prove that an n dimensional star has this property and it keeps this strong property even if there exist n 3 missing edges in it

hyper hamiltonian laceability on edge fault star graph the star graph possess many nice topological properties edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes in this paper we show that the n dimensional star graph is n 3 edge fault tolerant hamiltonian laceable n 3 edge fault tolerant strongly hamiltonian laceable and n 4 edge fault tolerant hyper hamiltonian laceable all these results are optimal in a sense described in this paper

routing properties of supercubes abstract usually each vertex of the s 1 dimensional hypercube is labeled with a unique integer k with 0 k 2 s 1 1 the supercube s n of n nodes with 2 s n 2 s 1 is constructed by merging nodes u and u 2 s with n u 2 s 1 1 in the s 1 dimensional hypercube into a single node labeled as u 2 s and leaving other nodes in the s 1 dimensional hypercube unchanged in this paper we give the exact distance between any two nodes of supercube and present a new shortest path routing algorithm on s n then we show how to construct s n disjoint paths between any two nodes of the supercube where s n is the connectivity of s n finally we compute the wide diameter and the fault diameter of s n we show that both the wide diameter and the fault diameter are equal to s 2 if n 2 s 1 2 i 1 0 i s 1 and s 1 otherwise

rg conditional diagnosability a novel generalized measure of system level diagnosis abstract system level diagnosis has become an important diagnosis method for multiprocessor systems among all system level diagnosis measures diagnosability is relatively small the conditional diagnosability constraint that each vertex has at least one good neighbor is relatively conservative when the dimension is far greater than 1 and g good neighbor conditional diagnosability does not consider this restriction on faulty vertices therefore a thorough study of diagnosability under the condition that each vertex has at least g good neighbors is an appealing subject motivated by r g vertex connectivity in this paper we introduce a novel generalized system level diagnosis measure named r g conditional diagnosability which assumes that every processor has at least g good neighbors the popular conditional diagnosability is a special case of r g conditional diagnosability when g 1 then we determine that the r g conditional diagnosability of n dimensional hypercube q n under the preparata metze chien pmc model is 2 2 g n 2 g 2 2 g 1 1

the g good neighbor conditional diagnosability of twisted hypercubes under the pmc and mm model abstract connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the rg vertex connectivity of a connected graph g is the minimum cardinality of a g good neighbor conditional faulty set x v g such that g x is disconnected the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify the twisted hypercube is a new variant of hypercubes with asymptotically optimal diameter introduced by x d zhu in this paper we first determine the rg vertex connectivity of twisted hypercubes then establish the g good neighbor conditional diagnosability of twisted hypercubes under the pmc model and mm model respectively

embedding paths and cycles in 3 ary n cubes with faulty nodes and links the k ary n cube denoted by q n k is one of the most important interconnection networks for parallel computing in this paper we consider the problem of embedding cycles and paths into faulty 3 ary n cubes let f be a set of faulty nodes and or edges and n 2 we show that when f 2n 2 there exists a cycle of any length from 3 to v q n 3 f in q n 3 f we also prove that when f 2n 3 there exists a path of any length from 2n 1 to v q n 3 f 1 between two arbitrary nodes in q n 3 f since the k ary n cube is regular of degree 2n the fault tolerant degrees 2n 2 and 2n 3 are optimal

edge disjoint paths in hypercubes and folded hypercubes with conditional faults it is known that edge disjoint paths is closely related to the edge connectivity and the multicommodity flow problems in this paper we study the edge disjoint paths in hypercubes and folded hypercubes with edge faults we first introduce the f strongly menger edge connectivity of a graph and we show that in all n dimensional hypercubes folded hypercubes respectively with at most 2 n 4 2 n 2 respectively edges removed if each vertex has at least two fault free adjacent vertices then every pair of vertices u and v are connected by min deg u deg v edge disjoint paths where deg u and deg v are the remaining degree of vertices u and v respectively

on conditional diagnosability of the folded hypercubes the n dimensional folded hypercube fq n a variation of the hypercube proposed by ahmed et al a el amawy s latifi properties and performance of folded hypercubes ieee transactions on parallel and distributed systems 2 3 1991 31 42 is an n 1 regular n 1 connected graph conditional diagnosability a new measure of diagnosability introduced by lai et al pao lien lai jimmy j m tan chien ping chuang lih hsing hsu conditional diagnosability measures for large multiprocessor systems ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of fq n t c fq n is 4n 3 when n 5 or n 8 t c fq 3 3 t c fq 4 7

on fault identification in diagnosable systems this paper begins by giving a characterization of t1 t1 diagnosable systems then a class of t0 diagnosable systems denoted by d n t0 x is considered it is shown for any member of this class that 1 necessary and sufficient conditions for t1 t1 diagnosability are greatly simplified 2 optimal diagnosis algorithms of time complexity 0 nt0 exist and most importantly 3 given the test results any set f of faults with f t1 can be identified to within a set f with f f and f min t1 f 1

long paths and cycles in hypercubes with faulty vertices a fault free path in the n dimensional hypercube q n with f faulty vertices is said to be long if it has length at least 2 n 2f 2 similarly a fault free cycle in q n is long if it has length at least 2 n 2f if all faulty vertices are from the same bipartite class of q n such length is the best possible we show that for every set of at most 2n 4 faulty vertices in q n and every two fault free vertices u and v satisfying a simple necessary condition on neighbors of u and v there exists a long fault free path between u and v this number of faulty vertices is tight and improves the previously known results furthermore we show for every set of at most n 2 10 n 2 1 faulty vertices in q n where n 15 that q n has a long fault free cycle this is a first quadratic bound which is known to be asymptotically optimal

fault diagnosability of arrangement graphs the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and to replace the faulty processors to maintain a system s high reliability the fault diagnosis is the process of identifying faulty processors in a system through testing in this paper we show that the largest connected component of the survival graph contains almost all of the remaining vertices in the n k arrangement graph a n k when the number of moved faulty vertices is up to twice or three times the traditional connectivity using this fault resiliency we establish the conditional diagnosability of a n k under the comparison model and prove that the conditional diagnosability of a n k is 3k 2 n k 3 for k 4 n k 2 and the conditional diagnosability of a n n 1 is 3n 7 for n 5

smooth linearization of commuting circle diffeomorphisms we show that a finite number of commuting diffeomorphisms with simultane ously diophantine rotation numbers are smoothly conjugated to rotations this solves a problem raised by moser

on reliability of the folded hypercubes in terms of the extra edge connectivity for a graph g and a non negative integer g the g extra edge connectivity of g is the minimum cardinality of a set of edges in g if it exists whose deletion disconnects g and each remaining component will have at least g vertices the extra edge connectivity is an important parameters for the reliability evaluation of interconnection networks in this paper we explore g extra edge connectivity g fq n of the folded hypercube fq n for g n denote g by i 0 s 2 t i where t 0 log 2 g and t i log 2 g r 0 i 1 2 t r we show that g fq n g n 1 i 0 s t i 2 t i i 0 s 2 i 2 t i for n 6 this result generalizes the previous results by zhu et al 2007 for 3 fq n and by hsieh and tsai in press for 4 fq n and so on

a kind of conditional vertex connectivity of cayley graphs generated by 2 trees let g be a graph then t v g is called an r k vertex cut if g t is disconnected and each vertex in v g t has at least k neighbors in g t the size of a smallest r k vertex cut is the r k vertex connectivity of g and is denoted by k k g in this paper we determine the numbers k 1 and k 2 for cayley graphs generated by 2 trees including the popular alternating group graphs

path embeddings in faulty 3 ary n cubes the class of k ary n cubes represents the most commonly used interconnection topology for distributed memory parallel systems in this paper we study the problem of embedding paths of various lengths into faulty 3 ary n cubes and prove that a faulty 3 ary n cube with f 2n 3 faulty vertices admits a path of every length from 2n 1 to 3 n f 1 connecting any two distinct healthy vertices this result is optimal with respect to the number of vertex faults tolerated

many to many disjoint paths in faulty hypercubes this paper considers the problem of many to many disjoint paths in the hypercube q n with f v faulty vertices and f e faulty edges and obtains the following result for any integer k with 1 k n 1 any two sets s and t of k fault free vertices in different parts if f v f e n k 1 then there exist k disjoint fault free s t paths in q n which contains at least 2 n 2 f v vertices this result is optimal in the worst case

a general technique to establish the asymptotic conditional diagnosability of interconnection networks we develop a general and demonstrably widely applicable technique for determining the asymptotic conditional diagnosability of interconnection networks prevalent within parallel computing under the comparison diagnosis model we apply our technique to replicate yet extend existing results for hypercubes and k ary n cubes before going on to obtain new results as regards folded hypercubes pancake graphs and augmented cubes in particular we show that the asymptotic conditional diagnosability of folded hypercubes fq n is 3n 2 pancake graphs p n is 3n 7 and augmented cubes aq n is 6n 17 we demonstrate how our technique is independent of structural properties of the interconnection network g in question and essentially only dependent upon the minimal size of the neighbourhood of a path of length 2 in g the number of neighbours any two distinct vertices of g have in common and the minimal degree of any vertex in g

on the genus of star graphs the star graph has recently been suggested as an alternative to the hypercube the star graph has a rich structure and symmetry properties as well as desirable fault tolerant characteristics the star graph s maximum vertex degree and diameter viewed as functions of network size grow less rapidly than the corresponding measures in a hypercube we investigate the genus of the star graph and compare it with the genus of the hypercube

conditional diagnosability of optical multi mesh hypercube networks under the comparison diagnosis model abstract due to integrated positive features of both hypercubes and tori optical multi mesh hypercube ommh networks are regarded as a class of promising optical interconnection topologies the notion of conditional diagnosability helps enhance the self diagnosing capability of multicomputers this paper determines the conditional diagnosabilities of ommh networks under the maeng malek comparison model

on local rigidity of partially hyperbolic affine ℤ k actions the following dichotomy for affine z k actions on the torus t d k d is an element of n is shown to hold i the linear part of the action has no rank one factors and then the affine action is

on 3 extra connectivity and 3 extra edge connectivity of folded hypercubes given a graph mbi g and a non negative integer g the g extra connectivity resp g extra edge connectivity of mbi g is the minimum cardinality of a set of vertices resp edges in mbi g if it exists whose deletion disconnects mbi g and leaves each remaining component with more than g vertices this study shows that the 3 extra connectivity resp 3 extra edge connectivity of an mbi n dimensional folded hypercube is 4 n 5 for n geq 6 resp 4 n 4 for n geq 5 this study also provides an upper bound for the g extra connectivity on folded hypercubes for g geq 6

component connectivity of hypercubes the r component connectivity c r g of a non complete graph g is the minimum number of vertices whose deletion results in a graph with at least r components in this paper we determine the component connectivity of the hypercube c r 1 q n r 2 2 2 n 5 2 r n 2 2 n 1 for n 1 r 2 n 5 n 6 this paper extends the results in hsu et al 2012 3

conditional edge fault tolerant edge bipancyclicity of hypercubes the hypercube has been one of the most popular interconnection networks for parallel computer communication systems in this paper we assume that each node is incident with at least two fault free links under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 6 to 2 n inclusive even if it has up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated

on c2 diffeomorphisms of the circle which are of type iii1 for every irrational number 0 1 which is not of constant type we construct ac 2 diffeomorphism of the circle with rotation number which is of type iii1 this diffeomorphism can be chosen arbitrarily close to the rotationr our methods also allow us to construct for every liouville number 0 1 ac diffeomorphism of the circle with rotation number which is of type iii1

on g good neighbor conditional diagnosability of n k star networks abstract the g good neighbor conditional diagnosability is a new measure for fault diagnosis of systems xu et al 2017 27 determined the g good neighbor conditional diagnosability of n k star networks s n k i e t g s n k with 1 k n 1 for 1 g n k under the pmc model and the mm model in this paper we determine t g s n k for all the remaining cases with 1 k n 1 for 1 g n 1 under the two models from which we can obtain the g good neighbor conditional diagnosability of the star graph obtained by li et al 2017 16 for 1 g n 2

super rk vertex connectedness abstract for a graph g v e a subset f v g is called an rk vertex cut of g if g f is disconnected and each vertex u v g f has at least k neighbours in g f the rk vertex connectivity of g denoted by k g is the cardinality of a minimum rk vertex cut of g in this paper we further study the rk vertex connectivity by introducing the concept called super rk vertex connectedness the graph g is called super rk vertex connectedness if for every minimum rk vertex cut s g s contains a component which is isomorphic to a certain graph h where h is related to the graph g and integer k for the cayley graphs generated by wheel graphs h is isomorphic to k2 when k 1 and h is isomorphic to c4 when k 2 in this paper we show that the cayley graphs generated by wheel graphs are super r1 vertex connectedness and super r2 vertex connectedness our studies generalize the main result in 8

relating diagnosability strong diagnosability and conditional diagnosability of strong networks an interconnection network s diagnosability is an important measure of its self diagnostic capability based on the classical notion of diagnosability strong diagnosability and conditional diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we study a class of interconnection networks called strong networks which are n regular n 1 connected and with cn number no more than n 3 we build a relationship among the three diagnosability measures for strong networks under both pmc and rm mm ast models given a strong network g with diagnosability t we prove that g is strongly t diagnosable if and only if g s conditional diagnosability is greater than t a simple check can show that almost all well known regular interconnection networks are strong networks the significance of this paper s result is that it reveals an important relationship between strong and conditional diagnosabilities and the proof of strong diagnosability for many interconnection networks under rm mm ast or pmc model is not necessary if their conditional diagnosability can be shown to be strictly larger than their diagnosability

strongly menger edge connectedness and strongly menger vertex connectedness of regular networks abstract let f be a conditional faulty edge set of a graph g here the condition imposed on f is g f r for some fixed integer r the graph g is called f strongly menger edge connected of order r if each pair of distinct vertices u and v are connected by min deg g f u deg g f v edge disjoint paths in g f where deg g f u and deg g f v are the degrees of u and v in g f respectively a graph g is t strongly menger edge connected of order r if g is f strongly menger edge connected of order r for every f e g with f t one can consider the vertex version in an analogous way there are a number of papers with results for r 2 for several popular classes of interconnection network in this paper we develop a unified approach by considering various sufficient conditions of a regular graph to be f strongly menger edge connected of order r and the corresponding vertex version as corollaries several results on fundamental classes of interconnection networks are presented including new results as well as strengthening of existing results such as the one given by qiao and yang 2017 20

restricted connectivity for three families of interconnection networks vertex connectivity and edge connectivity are two important parameters in interconnection networks even though they reflect the fault tolerance correctly they undervalue the resilience of large networks by the concept of conditional connectivity and super connectivity the concept of restricted vertex connectivity and restricted edge connectivity of graphs was proposed by esfahanian a h esfahanian generalized measures of fault tolerance with application to n cube networks ieee transactions on computers 38 1989 1586 1591 such measures take the resilience of large networks into consideration in this paper we propose three families of interconnection networks and discuss their restricted vertex connectivity and restricted edge connectivity in particular the hypercubes twisted cubes crossed cubes mobius cubes star graphs pancake graphs recursive circulant graphs and k ary n cubes are special cases of these families

diagnosable evaluation of dcc linear congruential graphs under the pmc diagnostic model a dcc disjoint consecutive cycles linear congruential graph g f n consists of n nodes and is generated by a set of linear functions f with special properties it was proved that g f n is a 2t regular graph and has connectivity 2t where t f and 1 3 and 2

the conditional fault diagnosability of n k star graphs abstract the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and replace the faulty processors to maintain a system s high reliability the fault diagnosis is the process of identifying faulty processors in a system through testing this paper shows that the largest connected component of the survival graph contains almost all the remaining vertices when a lot of faulty vertices occur in the n k star graph s n k n 5 k 3 n k 2 based on this fault resiliency it establishes the conditional fault diagnosability of s n k under the comparison model

adaptive system level diagnosis for hypercube multiprocessors using a comparison model fault detection and localization are important in multiprocessor computing systems several methods of system self diagnosis have been developed including the model proposed by preparata metze and chien known as the pmc model and a comparison model proposed by maeng and malek the mm model adaptive diagnosis is a practical system level diagnostic scheme whose main design objectives are to reduce the number of test rounds and the total number of tests a number of interesting research results have been presented for adaptive diagnosis using the classical pmc model however adaptive diagnosis using the mm model has been discussed only in relation to completely connected systems and torus systems in this paper we consider the problem of adaptive fault diagnosis in an n dimensional hypercube using the mm model the hypercube is an important interconnection network and an n dimensional hypercube has been shown to be n diagnosable using the mm model the goal is to correctly identify the status of all processors assuming that the number of faulty vertices does not exceed the hypercube dimension for a cycle of n vertices we show that at least four test rounds are necessary for complete diagnosis if n is not a multiple of three we propose a scheme that completely diagnoses a 4 dimensional hypercube in at most seven test rounds and an n dimensional hypercube for n 5 in at most six test rounds and at most n 2n 3 8n 2 tests where n 2 n

conditional diagnosability of matching composition networks under the mm model diagnosability is a critical metric for determining the reliability of a multiprocessor system in 2005 lai et al proposed a new measure for the fault diagnosis of a system i e conditional diagnosability in which it is assumed that at least one of the neighbors of an arbitrary node in the system is not faulty in this paper we obtain a sufficient condition for a class of networks called matching composition networks mcns which are conditionally t diagnosable under the mm model then we apply the sufficient condition to show the conditional diagnosability of bijective connection bc networks finally we show that the sufficient condition can be applied to networks other than bc networks

fault free hamiltonian cycles in twisted cubes with conditional link faults the n dimensional twisted cube denoted by tq n a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each vertex is incident with at least two fault free links we show that tq n can tolerate up to 2n 5 edge faults while retaining a fault free hamiltonian cycle the result is optimal with respect to the number of edge faults tolerated

constructing one to many disjoint paths in folded hypercubes routing functions have been shown effective in deriving disjoint paths in the hypercube in this paper using a minimal routing function k 1 disjoint paths from one node to another k 1 distinct nodes are constructed in a k dimensional folded hypercube whose maximal length is not greater than the diameter plus one which is minimum in the worst case for the general case the maximal length is nearly optimal spl les the maximal distance between the two end nodes of these k 1 paths plus two as a by product the rabin number of the folded hypercube is obtained which is an open problem raised by s c liaw and g j chang 1999

rigidity of some abelian by cyclic solvable group actions on mathbb t n in this paper we study a natural class of groups that act as affine transformations of mathbb t n we investigate whether these solvable abelian by cyclic groups can act smoothly and nonaffinely on mathbb t n while remaining homotopic to the affine actions in the affine actions elliptic and hyperbolic dynamics coexist forcing a priori complicated dynamics in nonaffine perturbations we first show using the kam method that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action provided certain diophantine conditions on the action are met in dimension two under natural dynamical hypotheses we get a complete classification of such actions namely any such group action by c r diffeomorphisms can be conjugated to the affine action by c r varepsilon conjugacy next we show that in any dimension c 1 small perturbations can be conjugated to an affine action via c 1 varepsilon conjugacy the method is a generalization of the herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics

conditional matching preclusion sets abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex in this paper we look for obstruction sets beyond these sets we introduce the conditional matching preclusion number of a graph it is the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings we find this number and classify all optimal sets for several basic classes of graphs

the g good neighbor conditional diagnosability of star graphs under the pmc and mm model diagnosability plays an important role in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify this paper shows that the g good neighbor conditional diagnosabilities of star graph sn under the pmc and mm model are ng g 1 1 for n4 and 0gn2 our study shows the g good neighbor conditional diagnosability of star graph is several times larger than the classical diagnosability of star graph

an 0 n 2 5 fault identification algorithm for diagnosable systems consider a system composed of n independent processors each of which tests a subset of the others it is assumed that at most tp of these processors are permanently faulty and that the outcome of a test is reliable if and only if the processor which performed the test is fault free such a system is said to be tp diagnosable if given any complete collection of test results the set of faulty processors can be uniquely identified in this paper it is shown that tp diagnosable systems due to their robust interconnection structure possess heretofore unknown graph theoretic properties relative to vertex cover sets and maximum matchings an 0 n2 5 algorithm is given which exploits these properties to identify the set of faulty processors in a tp diagnosable system the algorithm is shown to be correct complete not based on any conjecture and superior to any other known fault identification algorithm for the general class of tp diagnosable systems

some problems and results in the study of nonlinear analysis this paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis which were obtained by the author and some chinese mathematicians

strong rabin numbers of folded hypercubes the strong rabin number of a network w of connectivity k is the minimum l so that for any k 1 nodes s d1 d2 dk of w there exist k node disjoint paths from s to d1 d2 dk respectively whose maximal length is not greater than l where s d1 d2 dk and d1 d2 dk are not necessarily distinct in this paper we show that the strong rabin number of a k dimensional folded hypercube is k 2 1 where k 2 is the diameter of the k dimensional folded hypercube each node disjoint path we obtain has length not greater than the distance between the two end nodes plus two this paper solves an open problem raised by liaw and chang

the triangular pyramid routing and topological properties in this paper a new topology for multicomputer interconnection networks based on triangular mesh is proposed the new network referred to as the triangular pyramid or tripy for short has l levels of triangular mesh we study some basic important properties of the proposed network as well as introduce a routing algorithm for the tripy network based on the routing of triangular meshes we prove that this form of pyramidal network is hamiltonian hamiltonian connected and pancyclic we also prove that the proposed network is 6 colorable and conduct a brief comparison of the tripy and its traditional pyramid counterpart our results show that the proposed network has higher scalability connectivity and total network bandwidth while preserving the important properties of the traditional pyramid network

conditional diagnosability of arrangement graphs under the pmc model abstract processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault set can t contain all the neighbors of any node in a multiprocessor system this paper considers the conditional diagnosability of an n k arrangement graph a n k a flexible interconnection network model for multiprocessor systems under the classical pmc diagnostic model and determines that the conditional diagnosability of a n k k 2 n k 2 is 4 k 4 n k 3 which is about four times its traditional diagnosability

a new proof of m herman s theorem a new proof of the m herman theorem on the smooth conjugacy of a circle map is presented here it is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables

on reliability of the folded hypercubes in this paper we explore the 2 extra connectivity and 2 extra edge connectivity of the folded hypercube fq n we show that k 2 fq n 3n 2 for n 8 and l 2 fq n 3n 1 for n 5 that is for n 8 resp n 5 at least 3n 2 vertices resp 3n 1 edges of fq n are removed to get a disconnected graph that contains no isolated vertices resp edges when the folded hypercube is used to model the topological structure of a large scale parallel processing system these results can provide more accurate measurements for reliability and fault tolerance of the system

the 1 2 good neighbor conditional diagnosabilities of regular graphs abstract fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems in 2012 peng et al proposed a new measure for the fault diagnosis of systems namely g good neighbor conditional diagnosability which requires that any fault free vertex has at least g fault free neighbors in the system the g good neighbor conditional diagnosabilities of a graph g under the pmc model and the mm model are denoted by t g pmc g and t g mm g respectively in this paper we first determine that t g pmc g t g mm g if g 2 second we establish a general result on the 1 2 good neighbor conditional diagnosabilities of some regular graphs as applications the 1 2 good neighbor conditional diagnosabilities of bc graphs folded hypercubes and four classes of cayley graphs namely unicyclic transposition graphs wheel transposition graphs complete transposition graphs and tree transposition graphs are determined under the pmc model and the mm model in addition we determine the r2 connectivities of bc graphs and folded hypercubes

long paths in hypercubes with conditional node faults let f be a set of f 2 n 5 faulty nodes in an n cube q n such that every node of q n still has at least two fault free neighbors then we show that q n f contains a path of length at least 2 n 2 f 1 respectively 2 n 2 f 2 between any two nodes of odd respectively even distance since the n cube is bipartite the path of length 2 n 2 f 1 or 2 n 2 f 2 turns out to be the longest if all faulty nodes belong to the same partite set as a contribution our study improves upon the previous result presented by j s fu longest fault free paths in hypercubes with vertex faults information sciences 176 2006 759 771 where only n 2 faulty nodes are considered

orienting cayley graphs generated by transposition trees day and tripathi k day a tripathi unidirectional star graphs inform process lett 45 1993 123 129 proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs cheng and lipman e cheng m j lipman on the day tripathi orientation of the star graphs connectivity inform process lett 73 2000 5 10 e cheng m j lipman connectivity properties of unidirectional star graphs congr numer 150 2001 33 42 studied the connectivity properties of these unidirectional star graphs the class of star graphs is a special case of cayley graphs generated by transposition trees in this paper we give directions on these graphs and study the connectivity properties of the resulting unidirectional graphs

pancyclicity and bipancyclicity of conditional faulty folded hypercubes a graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order the pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network s topology can simulate rings of various lengths an n dimensional folded hypercube fq n is an attractive variant of an n dimensional hypercube q n that is obtained by establishing some extra edges between the vertices of q n fq n for any odd n is known to be bipartite in this paper we explore the pancyclicity and bipancyclicity of fq n for any fq n n 2 with at most 2n 3 faulty edges where each vertex is incident to at least two fault free edges we prove that there exists a fault free cycle of every even length from 4 to 2 n and when n 2 is even we prove there also exists a fault free cycle of every odd length from n 1 to 2 n 1 the result is optimal with respect to the number of faulty edges tolerated

the 2 good neighbor diagnosability of cayley graphs generated by transposition trees under the pmc model and mm model diagnosability is an important metric for measuring the reliability of multiprocessor systems in 2012 peng et al proposed a new measure for fault diagnosis of the system which is called g good neighbor diagnosability that restrains every fault free node containing at least g fault free neighbors as a favorable topology structure of interconnection networks the cayley graph c n generated by the transposition tree n has many good properties in this paper we give that the 2 good neighbor diagnosability of c n under the pmc model and mm model is g n 2 1 where n 4 and g is the girth of c n

existence theorems of solutions for generalized quasi variational inequalities a minimax theorem and a section theorem in the spaces without linear structure in this paper by using particular techniques two existence theorems of solutions for generalized quasi variational inequalities a minimax theorem and a section theorem in the spaces without linear structure are established and finally a new coincidence theorem in locally convex spaces is obtained

conditional diagnosability and strong diagnosability of split star networks under the pmc model an interconnection network s diagnosability is an important measure of its self diagnostic capability the classical problems of fault diagnosis are explored widely the conditional diagnosability is proposed by lai et al as a new measure of diagnosability which can better measure the diagnosability of regular interconnection networks the conditional diagnosability is an important indicator of the robustness of a multiprocessor system in presence of failed processors furthermore a multiprocessor system is strongly t diagnos able if it is t diagnosable and can achieve diagnosability t 1 except for the case where a node s neighbors are all faulty the conditional diagnosability and strong diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we determine the conditional diagnosability of an n dimensional split star network denoted as s n 2 a well known interconnection network model for multiprocessor systems under the pmc preparata metze and chien model we show that the conditional diagnosability of s n 2 n 4 is 8 n 23 which is about four times of its traditional diagnosability as a byproduct the strong diagnosability of s n 2 is also obtained we investigate the combinatorial properties and fault tolerant properties of the split star network we show that as long as a fault set f has 6 n 17 or fewer nodes s n 2 f excluding a subset of at most two nodes contains one large connected component we show that as long as a fault set f has 8 n 25 or fewer nodes s n 2 f excluding a subset of at most three nodes contains one large connected component we show that the classic diagnosability of s n 2 n 2 under the pmc model is 2 n 3 we also establish that the conditional diagnosability and the strong diagnosability of s n 2 are 8 n 23 and 2 n 3 under the pmc model

on conditional fault tolerance and diagnosability of hierarchical cubic networks fault tolerance is especially important for interconnection networks since the growing size of networks increases their vulnerability to component failures a classical measure for the fault tolerance of a network in the case of vertex failures is its connectivity given a network based on a graph g and a positive integer h the r h connectivity of g is the minimum cardinality of a set of vertices in g if any whose deletion disconnects g and the minimum degree of every connected component is at least h this paper investigates the r h connectivity h 1 2 of the hierarchical cubic network hcn n n 2 and shows that 1 hcn n 2 n 2 hcn n 4 n 4 respectively furthermore the paper establishes the conditional diagnosability of hcn n under the pmc diagnostic model

linearly many faults in dual cube like networks the dual cubes were introduced as a better interconnection network than the hypercubes for large scale distributed memory multiprocessors in this paper we introduce a generalization of these networks called dual cube like networks which preserve the basic structure of dual cubes and retain many of its topological properties we investigate structural properties of these networks beyond simple measures such as connectivity we prove that if up to kn k k 1 2 vertices are deleted from a dual cube like n regular network then the resulting graph will either be connected or will have a large component and small components having at most k 1 vertices in total and this result is sharp for k n as an application we derive additional results such as the cyclic vertex connectivity and the restricted vertex connectivity of these networks

linear array and ring embeddings in conditional faulty hypercubes the n dimensional hypercube qn is a graph having 2n vertices labeled from 0 to 2n 1 two vertices are connected by an edge if their binary labels differ in exactly one bit position in this paper we consider the faulty hypercube qn with n 3 that each vertex of qn is incident to at least two nonfaulty edges based on this requirement we prove that qn contains a hamiltonian path joining any two different colored vertices even if it has up to 2n 5 edge faults moreover we show that there exists a path of length 2n 2 between any two the same colored vertices in this faulty qn furthermore we also prove that the faulty qn still contains a cycle of every even length from 4 to 2n inclusive

fault tolerance and diagnosability of burnt pancake networks under the comparison model processor fault diagnosis plays an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a new measure of diagnosability by assuming an additional condition that any faulty set can t contain all the neighbors of any node in a multiprocessor system in this paper we explore algebraic and combinatorial properties of burnt pancake networks and investigate the structural vulnerability as well as super and extra connectivities furthermore we show that the classic diagnosability and the conditional diagnosability of n dimensional burnt pancake network bp n n 4 are n and 3 n 4 respectively algebraic and combinatorial properties of burnt pancake networks are established the structural vulnerability as well as super and extra connectivities are explored the classic and conditional diagnosabilities under the comparison model are derived

reliability evaluation of bc networks in terms of the extra vertex and edge connectivity reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems the extra connectivity and the extra edge connectivity are two important parameters for the reliability evaluation of interconnection networks the mbi n dimensional bijective connection network in brief bc network includes several well known network models such as hypercubes mobius cubes crossed cubes and twisted cubes in this paper we explore the extra connectivity and the extra edge connectivity of bc networks and discuss the structure of bc networks with many faults we obtain a sharp lower bound of g extra edge connectivity of an mbi n dimensional bc network for n geq 4 and 1 leq g leq 2 n over 2 we also obtain a sharp lower bound of g extra connectivity of an n dimensional bc network for n geq 4 and 1 leq g leq 2 n which improves the result in reliability evaluation of bc networks ieee trans computers doi 10 1109 tc 2012 106 for 1 leq g leq n 3 furthermore we give a remark about exploring the g extra edge connectivity of bc networks for the more general mbi g and we also characterize the structure of bc networks with many faulty nodes or links as an application we obtain several results on the mbi g extra edge connectivity and the structure of faulty networks on hypercubes mobius cubes crossed cubes and twisted cubes

section theorems on h spaces with applications abstract some further generalizations of fan s and shih tan s geometric theorems for convex sets are obtained as applications in sections 3 5 we utilize these results to study the minimax inequalities and the fixed point theorems for multi valued mappings the results presented in this paper generalize the von neumann and the fan kneser minimax theorems and some others recent results

the t k diagnosability of star graph networks the t k diagnosis is a diagnostic strategy at system level that can significantly enhance the system s self diagnosing capability it can detect up to t faulty processors or nodes units which might include at most k misdiagnosed processors where k is typically a small number somani and peleg 1996 claimed that an n dimensional star graph denoted s n a well studied interconnection model for multiprocessor systems is k 1 n 3k 2 k diagnosable recently chen and liu 2012 found counterexamples for the diagnosability obtained in without further pursuing the cause of the flawed result in this paper we provide a new complete proof that an mbi n dimensional star graph is actually k 1 n 3k 1 k diagnosable where 1 leq k leq 3 and investigate the reason that caused the flawed result in based on our newly obtained fault tolerance properties we will also outline an o n log n diagnostic algorithm n n is the number of nodes in s n to locate all up to k 1 n 3k 1 faulty processors among which at most k 1 leq k leq 3 fault free processors might be wrongly diagnosed as faulty

longest fault free paths in star graphs with vertex faults the star graph sn has been recognized as an attractive alternative to the hypercube since s1 s2 and s3 have trivial structures we focus our attention on sn with n 4 in this paper let fv denote the set of faulty vertices in sn we show that when fv n 5 sn with n 6 contains a fault free path of length n 2 fv 2 n 2 fv 1 between arbitrary two vertices of even odd distance since sn is bipartite with two partite sets of equal size the path is longest for the worst case scenario the situation of n 4 and fv n 5 is also discussed

diagnosability of hypercubes and enhanced hypercubes under the comparison diagnosis model a sengupta and a dahbura 1992 discussed how to characterize a diagnosable system under the comparison diagnosis model proposed by j maeng and m malek 1981 and a polynomial algorithm was given to identify the faulty processors provided that the system s diagnosability is known however for a general system the determination of its diagnosability is not algorithmically easy this paper proves that for the important hypercube structured multiprocessor systems n cubes the diagnosability under the comparison model is n when n spl ges 5 the paper also studies the diagnosability of enhanced hypercube which is obtained by adding 2 sup n 1 more links to a regular hypercube of 2 sup n processors it is shown that the augmented communication ability among processors also increases the system s diagnosability under the comparison model we prove that the diagnosability is n 1 for an enhanced hypercube when n spl ges 6

conditional diagnosability of alternating group graphs let an be the alternating group of degree n with n 3 set s 1 2 i 1 i 2 3 i n the alternating group graph denoted by agn is defined as the cayley graph on an with respect to s jwo et al networks 23 1993 315 326 introduced alternating group graph agn as an interconnection network topology for computing systems conditional diagnosability a new measure of diagnosability introduced by lai et al ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of agn is 4 for n 4 and 6n 18 for each n 5

long paths in hypercubes with a quadratic number of faults a path between distinct vertices u and v of the n dimensional hypercube q n avoiding a given set of f faulty vertices is called long if its length is at least 2 n 2f 2 we present a function f n email protected n 2 such that if f f n then there is a long fault free path between every pair of distinct vertices of the largest fault free block of q n moreover the bound provided by f n is asymptotically optimal furthermore we show that assuming f f n the existence of a long fault free path between an arbitrary pair of vertices may be verified in polynomial time with respect to n and if the path exists its construction performed in linear time with respect to its length

linearly many faults in cayley graphs generated by transposition trees abstract we prove that when linearly many vertices are deleted in a cayley graph generated by a transposition tree the resulting graph has a large connected component containing almost all remaining vertices

fault free hamiltonian cycles in crossed cubes with conditional link faults the crossed cube which is a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each node is incident with at least two fault free links we show that an n dimensional crossed cube contains a fault free hamiltonian cycle even if there are up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated we also verify that the assumption is practically meaningful by evaluating its occurrence probability which is very close to 1

the g good neighbor diagnosability of n k star graphs many large scale multiprocessor or multi computer systems take interconnection networks as underlying topologies fault diagnosis is especially important to identify fault tolerability of such systems the g good neighbor conditional diagnosability such that every fault free node has at least g fault free neighbors is a novel measure of diagnosability in this paper we show that the g good neighbor diagnosability of the n k star graph s n k under the pmc model 2 k n 1 and 1 g n k and the comparison model 2 k n 1 and 2 g n k is n g k 1 1 respectively in addition we derive that 1 good neighbor diagnosability of s n k under the comparison model is n k 2 for 3 k n 1 and n 4 as a supplement we also derive that the g good neighbor diagnosability of the n 1 star graph s n 1 1 g n 2 1 and n 4 under the pmc model and the comparison model is n 2 1 respectively we explore fault diagnosability of multiprocessor systems based on combinatorial network theory we establish the g good neighbor diagnosability of the n k star graph s n k under the pmc model we establish the g good neighbor diagnosability of the n k star graph s n k under the comparison model

reliability measures in relation to the h extra edge connectivity of folded hypercubes the folded hypercube fq n as a variation of the hypercube q n was proposed by a el amawy and s latifi in 1991 the h extra edge connectivity of the underlying topological graph of a multiprocessor system is a kind of measure for the reliability of the multiprocessor system in this paper we determine the exact value of h fq n for integer h 1 h 2 n 2 1 and 6 n which generalizes several known results for h n more interestingly we also show that h fq n is the constant n 2 r 1 2 n 2 r for 2 n 2 r l r h 2 n 2 r where r 1 2 n 2 1 and l r 2 2 r 1 3 if n is odd and l r 2 2 r 1 2 3 if n is even in particular for r n 2 1 2 n 2 3 h 2 n 1 h fq n 2 n

hamiltonian connectivity and strongly hamiltonian laceability of folded hypercubes in this paper we analyze a hypercube like structure called the folded hypercube which is basically a standard hypercube with some extra links established between its nodes we first show that the n dimensional folded hypercube is bipartite when n is odd we also show that the n dimensional folded hypercube is strongly hamiltonian laceable when n is odd and is hamiltonian connected when n 1 or n 2 is even

an approach to conditional diagnosability analysis under the pmc model and its application to torus networks a general technique is proposed for determining the conditional diagnosability of interconnection networks under the pmc model several graph invariants are involved in the approach such as the length of the shortest cycle the minimum number of neighbors p resp p over all p vertex subsets resp cycles and a variant of connectivity called the r super connectivity an n dimensional torus network is defined as a cartesian product of n cycles ck1 ckn where ckj is a cycle of length kj for 1 j n the proposed technique is applied to the two or higher dimensional torus networks and their conditional diagnosabilities are established completely the conditional diagnosability of every torus network g is equal to 4 g 1 excluding the three small ones c3 c3 c3 c4 and c4 c4 in addition p g as well as 4 g is derived for 2 p 4 and the r super connectivity is also derived for 1 r 3

on the conditional diagnosability of matching composition networks the conditional diagnosability of interconnection networks has been studied by using a number of ad hoc methods recently gathering various ad hoc methods developed in the last decade a unified approach was developed and this approach was used to find the conditional diagnosability of many interconnection networks in this paper we study the conditional diagnosability of matching composition networks including those that are not triangle free

linearly many faults in cayley graphs generated by transposition triangle free unicyclic graphs abstract we consider a kind of cayley graphs including the modified bubble sort graphs m b n the cayley graphs generated by lollipop graphs h n n 1 and the other cayley graphs generated by unicyclic triangle free graphs we get that if we delete linearly many vertices in cayley graphs generated by transposition unicyclic triangle free graphs then the resulting graphs has a large connected component that comprises of nearly all residual vertices

edge fault tolerance analysis of a class of interconnection networks fault tolerant measures have played an important role in the reliability of an interconnection network edge connectivity restricted edge connectivity extra edge connectivity and super edge connectivity of many well known interconnection networks have been explored in this paper we study the 2 extra edge connectivity of a special class of graphs g g0 g1 m proposed by chen et al appl math comput 140 2003 245 254 then by showing that several well known interconnection networks such as hyper

a kind of conditional connectivity of cayley graphs generated by unicyclic graphs abstract for a graph g v e a subset f v g is called an r k vertex cut of g if g f is disconnected and each vertex u v g f has at least k neighbors in g f the r k vertex connectivity of g denoted by k g is the cardinality of the minimum r k vertex cut of g which is a refined measure for the fault tolerance of network g in this paper we study 2 for cayley graphs generated by unicyclic graphs let sym n be the symmetric group on 1 2 n and t be a set of transpositions of sym n let g t be the graph on vertex set 1 2 n and edge set ij ij t if g t is a unicyclic graph then simply denote the cayley graph cay sym n t by ug n in particular if g t is a cycle then ug n is the well known modified bubble sort graph mb n we prove in this paper that if m is the length of the unique cycle in g t then 2 ug n 4 n 10 if 3 m n 4 n 8 if 4 m n

a kind of conditional connectivity of cayley graphs generated by wheel graphs for a connected graph g v e a subset fv is called an rk vertex cut of g if gf is disconnected and each vertex in vf has at least k neighbors in gf the cardinality of the minimum rk vertex cut is the rk vertex connectivity of g and is denoted by k g the conditional connectivity is a measure to explore the structure of networks beyond the vertex connectivity let sym n be the symmetric group on 1 2 n and t be a set of transpositions of sym n denote by g t the graph with vertex set 1 2 n and edge set ij ij t if g t is a wheel graph then simply denote the cayley graph cay sym n t by wgn in this paper we determine the values of 1 and 2 for cayley graphs generated by wheel graphs and prove that 1 wgn 4n6 and 2 wgn 8n18

a note on generalized matching preclusion in bipartite graphs abstract for a graph with an even number of vertices the matching preclusion number is the minimum number of edges whose deletion results in a graph with no perfect matchings the conditional matching preclusion number introduced as an extension of the matching preclusion number has the additional requirement that the resulting graph have no isolated vertices in bipartite graphs the edge sets that achieve these minimum numbers can be construed in terms of obstruction sets of vertices from this perspective the generalized matching preclusion problem was developed the matching preclusion number of order a is the minimum number of edges whose deletion results in a graph with no perfect matchings and no obstruction sets with fewer than a vertices in this paper we extend known sufficient conditions regarding the matching preclusion and conditional matching preclusion numbers orders 1 and 2 respectively to give sufficient conditions for the matching preclusion problem of order a for any positive integer a

conditional connectivity of star graph networks under embedding restriction the n dimensional star graph s n is one of the most attractive interconnection networks of large scale multiprocessor systems the k embedding restricted edge connectivity h k s n z k s n of s n is defined to be the cardinality of a minimum subset of edges nodes if any whose deletion disconnects s n and each node of the remaining components lies in an undamaged k dimensional substar s k in this paper we investigate the k embedding restricted edge connectivity of s n and determine the values of h k s n and z k s n for some k s

super connectivity and super edge connectivity for some interconnection networks let g v e be a k regular graph with connectivity k and edge connectivity l g is maximum connected if k k and g is maximum edge connected if l k moreover g is super connected if it is a complete graph or it is maximum connected and every minimum vertex cut is x v x e for some vertex email protected v and g is super edge connected if it is maximum edge connected and every minimum edge disconnecting set is v x v x e for some vertex email protected v in this paper we present three schemes for constructing graphs that are super connected and super edge connected applying these construction schemes we can easily discuss the super connected property and the super edge connected property of hypercubes twisted cubes crossed cubes mobius cubes split stars and recursive circulant graphs

embedding longest fault free paths onto star graphs with more vertex faults the n dimensional star graph sn belongs to a class of bipartite graphs and it is recognized as an attractive alternative to the hypercube since s1 s2 and s3 have trivial structures we focus our attention on sn with n 4 in this paper let f sn be the set of vertex faults previously it was shown that when f sn n 5 sn with n 6 can embed a longest fault free path of length at least n 2 f sn 1 respectively n 2 f sn 2 between two arbitrary vertices in different partite sets respectively the same partite set longest fault free paths in star graphs with vertex faults theoretical computer science 262 2001 215 227 in this paper we improve the above result by tolerating more faults f sn n 3 to embed a longest fault free path between two arbitrary vertices in sn n 4