Induced Subgraph Isomorphism: Are Some Patterns Substantially Easier Than Others?

Finding and Counting Patterns in Sparse Graphs

“…This theorem implies that all X-induced-minor-free, treewidth 2 patterns have O(m 2 )-time homomorphism counting algorithms. This is an improvement for sparse host graph even over the fast matrix multiplication based algorithm given by Curticapean, Dell, and Marx [9] for counting homomorphisms from treewidth 2 graphs that runs in O(n ω )-time. Since the spasm of P 10 does not contain any treewidth 4 graph or graph with an X-induced minor, we can show that there is an O(m 2 )-time algorithm for counting subgraph isormophisms of all paths on at most 10 vertices by showing that all treewidth 3 graphs in the spasm of P 10 has matched treewidth 3.…”

“…This problem seems to be much harder. It is conjectured by Floderus, Kowaluk, Lingas, Lundell [9] that counting induced subgraphs for any k-vertex pattern graph is as hard as counting k-cliques for sufficiently large k. Several works have considered the parameterized complexity of counting subgraphs (See [5,4,17,10,7]) where the primary goal is to obtain a dichotomy of easy vs hard based on structural graph parameters. Some works have also considered restrictions on host graphs such as d-degeneracy [3].…”