Generating all maximal induced subgraphs for hereditary and connected-hereditary graph properties

Abstract

This paper investigates a graph enumeration problem, called the maximal P -subgraphs problem, where P is a hereditary or connected-hereditary graph property. Formally, given a graph G, the maximal P -subgraphs problem is to generate all maximal induced subgraphs of G that satisfy P . This problem differs from the well-known node-deletion problem, studied by Yannakakis and Lewis [J. Lewis, On the complexity of the maximum subgraph problem, in: Proc. 10th Annual ACM Symposium on Theory of Computing, ACM Press, New York, USA, 1978, pp. 265–274; M. Yannakakis, Node- and edge-deletion NP-complete problems, in: Proc. 10th Annual ACM Symposium on Theory of Computing, ACM Press, New York, USA, 1978, pp. 253–264; J. Lewis, M. Yannakakis, The node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 (2) (1980) 219–230]. In the maximal P -subgraphs problem, the goal is to produce all (locally) maximal subgraphs of a graph that have property P , whereas in the node-deletion problem, the goal is to find a single (globally) maximum size subgraph with property P . Algorithms are presented that reduce the maximal P -subgraphs problem to an input-restricted version of this problem. These algorithms imply that when attempting to efficiently solve the maximal P -subgraphs problem for a specific P , it is sufficient to solve the restricted case. The main contributions of this paper are characterizations of when the maximal P -subgraphs problem is in a complexity class C (e.g., polynomial delay, total polynomial time).