Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph G whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of G having the same number of blue vertices and red vertices. They showed that this problem is NPhard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in O(n3) time for trees and O(n  m) time for split graphs and properly colored bipartite graphs, where n is the number of vertices and m is the number of edges.  this :111,paper, we show that BCS can be solved in O(n2) time for trees and O(n3) time for interval graphs. The former result can be extended to bounded treewidth graphs. We also consider a weighted version of BCS (WBCS). We prove that this variant is weakly NPhard even on star graphs and strongly NPhard even on split graphs and properly colored bipartite graphs, whereas the unweighted counterpart is tractable on those graph classes. Finally, we consider an exact exponentialtime algorithm for general graphs. We show that BCS can be solved in 2n2nO(1) time. This algorithm is based on a variant of DreyfusWagner algorithm for the Steiner tree problem.

Subset feedback vertex set on graphs of bounded independent set size Abstract   The ( Weighted )  Subset Feedback Vertex Set  problem is a generalization of the classical  Feedback Vertex Set  problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although  Subset Feedback Vertex Set  and  Feedback Vertex Set  exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that  Weighted Feedback Vertex Set  can be solved in polynomial time. We provide a dichotomy result with respect to the size xcexb1 of a maximum independent set. In particular we show that  Weighted Subset Feedback Vertex Set  can be solved in polynomial time for graphs with    xcexb1  xe2x89xa4  3   , whereas we prove that the problem remains NPhard for graphs with    xcexb1  xe2x89xa5  4   . Moreover, we show that the (unweighted)  Subset Feedback Vertex Set  problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time      n    O  (  xcexb1  )     . To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size.  Node Multiway Cut  is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for  Subset Feedback Vertex Set , we settle the complexity of  Node Multiway Cut  as well as its variants where nodes are weighted andor the terminals are deletable, for every value of the given independent set number.