Counting Subgraphs via Homomorphisms

Omid Amini, Fedor V. Fomin, Saket Saurabh
2012 SIAM Journal on Discrete Mathematics  
We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, Husfeldt and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings
more » ... of a bipartite graph, and color coding based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions and the color coding technique, we obtain the following new results: • The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time O(2 n+o(n) ); in particular in time O(2 n n 3 ) for trees and in time 2 n+O( √ n) for planar graphs. • Counting all maximum planar subgraphs, subgraphs of bounded genus, or more generally subgraphs excluding a fixed graph M as a minor can be done in 2 O(n) time. • Counting all subtrees with a given maximum degree (a generalization of counting Hamiltonian paths) of a given graph can be done in time 2 O(n) .
doi:10.1137/100789403 fatcat:igr4ivf2z5gg3h3zabvnwnfifm