From matchings to independent sets

Abstract

In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs.

Maximum independent sets in subcubic graphs: New results

“…• 6 does not belong to C . Indeed, if 6 belongs to C , then it must be of distance at most 2 from a (Claim 1), in which case we obtain an induced cycle of length at least p − 2 containing four edges (0, 3), (3,6), (0, 2), (2,5) • u is adjacent to 2. Indeed, assume u is not adjacent to 2.…”

“…If a is not adjacent to 3 and is adjacent to 6, then -if 6 ∈ C , i.e. if 6 = b, then 3 has no other neighbours on C by Claim 1, in which case we obtain an induced cycle of length p − 1 containing four edges (0, 3), (3,6), (0, 2), (2, 5) of H . -if 6 / ∈ C , then any other neighbour of 6 on C (if any) must be of distance at most 3 from a by Claim 1, in which case we obtain an induced cycle of length at least p − 2 containing four edges (0, 3), (3,6), (0, 2), (2,5) of H .…”

“…In other words, if the set M of forbidden induced subgraphs is finite, then the presence in M of a graph every connected component of which is a tripod is a necessary condition for polynomial-time solvability of the maximum independent set problem in the class of M-free subcubic graphs (assuming P = N P ). In [6], it was conjecture that this condition is also sufficient. Moreover, for graphs of bounded vertex degree the problem can be easily reduced to connected forbidden induced graphs, in which case the conjecture can be restated as follows.…”

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

“…Theorem 1 implies that (unless P = NP) for any graph F, if WIS can be solved for F-free graphs in polynomial time, then each component of F is S i;j;k for some indices i, j, k. Then Lozin [25] conjectured that WIS can be solved in polynomial time for S i;j;k -graphs for any fixed indices i, j, k. The above allows one to focus on possible open problems, i.e., on possible graph classes for which WIS may be solved in polynomial time.…”

Independent sets in ($P_4+P_4$,Triangle)-free graphs

“…Theorem 1 implies that (unless P = NP) for any graph F , if WIS can be solved for F -free graphs in polynomial time, then each connected component of F is S i,j,k for some indices i, j, k. Then Lozin [23] conjectured that WIS can be solved in polynomial time for S i,j,k -graphs for any fixed indices i, j, k. The above allows one to focus on possible open problems, i.e., on possible graph classes for which WIS may be solved in polynomial time. This manuscript shows that (i) WIS can be solved for (P 4 + P 4 , Triangle)-free graphs in polynomial time, and that in particular it turns out that (ii) for every (P 4 + P 4 , Triangle)-free graph G there is a family S of subsets of V (G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of S.…”