Abstract. In this paper we introduce a new simple strategy into edge-searching of a graph, which is useful to the various subgraph listing problems. Applying the strategy, we obtain the following four algorithms. The first one lists all the triangles in a graph G in O(a(G) All the algorithms require linear space. We also establish an upper bound on a(G) for a graph G: occurs. We will show in the succeeding section that the procedure above requires O(a(G)m) time. Throughout this paper m is the number of edges of a graph G, n is the number of vertices of G, and a(G) is the arboricity of G, that is, the minimum number of edge-disjoint spanning forests into which G can be decomposed [5] . We use the rather unfamiliar graph invariant a(G) as a parameter in bounding the running time of algorithms. The strategy yields simple algorithms for the problems to list certain kinds of subgraphs of a graph. The kinds of these subgraphs include "triangle, quadrangle," "complete subgraph of a fixed order," and "clique." Our algorithms are as fast as the known ones if any, and a factor n is often reduced to a(G) in the time complexity. In 2 we give an upper bound on a(G) for a general graph G: a(G) <-
Chiba and Nishizeki REF use a ranking algorithm that counts the total number of 4-cycles in a graph in O αm time, where α is the arboricity of the graph.
Arboricity and subgraph listing algorithms
SIAM J. Comput.
SIAM J. Comput.
Mathematics Computer Science