| Summary: | © 2019 Association for Computing Machinery. We consider the pattern detection problem in graphs: given a constant size pattern graph H and a host graph G, determine whether G contains a subgraph isomorphic to H. We present the following new improved upper and lower bounds: We prove that if a pattern H contains a k-clique subgraph, then detecting whether an n node host graph contains a not necessarily induced copy of H requires at least the time for detecting whether an n node graph contains a k-clique. The previous result of this nature required that H contains a k-clique which is disjoint from all other k-cliques of H. We show that if the famous Hadwiger conjecture from graph theory is true, then detecting whether an n node host graph contains a not necessarily induced copy of a pattern with chromatic number t requires at least the time for detecting whether an n node graph contains a t-clique. This implies that: (a) under Hadwiger’s conjecture for every k-node pattern H, finding an induced copy of H requires at least the time of k-clique detection and size ω(n k/4) for any constant depth circuit, and (b) unconditionally, detecting an induced copy of a random G(k, p) pattern w.h.p. requires at least the time of Θ(k/log k)-clique detection, and hence also at least size nΩ(k/log k) for circuits of constant depth. We show that for every k, there exists a k-node pattern that contains a k − 1-clique and that can be detected as an induced subgraph in n node graphs in the best known running time for k − 1-Clique detection. Previously such a result was only known for infinitely many k. Finally, we consider the case when the pattern is a directed cycle on k nodes, and we would like to detect whether a directed m-edge graph G contains a k-Cycle as a not necessarily induced subgraph. We resolve a 14 year old conjecture of [Yuster-Zwick SODA’04] on the complexity of k-Cycle detection by giving a tight analysis of their k-Cycle algorithm. Our analysis improves the best bounds for k-Cycle detection in directed graphs, for all k > 5.
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