Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

Abstract

We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in L 2 (G), the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given a graph class G characterized by a vertex ordering, and a graph G = (V, E) oe G with a corresponding vertex ordering ‡ of V , one can produce (in linear time in the size of G) an ordering on the vertices of L 2 (G), that shows that L 2 (G) oe G -for a number of graph classes G -without computing the line graph or the square of the line graph of G. These results generalize and unify previous ones on showing closure under L 2 (·) for various graph families. Furthermore, these orderings on L 2 (G) can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first O(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known O(n 4 ) time algorithm for the unweighted case.

Efficient algorithms for maximum induced matching problem in permutation and trapezoid graphs

“…With the combination of our algorithms and the current best graph identification algorithms that generate presentation models for permutation graphs and trapezoid graphs, we can solve the MIM problem in permutation and trapezoid graphs in linear and O(n 2 ) time, respectively. Our results are far better than the best known O(mn) algorithm [21] in both graph classes. This paper is the complete version including preliminary results of our conference papers [22] and [23].…”

“…However, to the best of our knowledge, the MIM problem has not been mentioned with specific algorithms on permutation graphs. The best-known algorithm for finding a weighted induced matching for all co-comparability graphs, which are a superclass of permutation graphs and trapezoid graphs, has time complexity of O(mn) [21]. So far, there is no other superclass of permutation graphs and trapezoid graphs on which the MIM problem is proved to be solvable in faster time than O(mn).…”

A new graph parameter to measure linearity

“…Given a graph class $${\mathscr{G}}$$, a VOC of $${\mathscr{G}}$$ is a characterization of a graph class given by the existence of a total ordering on the vertices with specific properties. VOCs have led to a number of efficient algorithms, and are often the basis of various graph recognition algorithms, see, for instance, [4, 5, 12, 14, 19]. In this section, we describe some of these VOCs for the graph classes for which we will prove the validity of Conjecture 1.2 in Section 4.…”

A New Graph Parameter to Measure Linearity

“…We call the triple a, b, c as described in Theorem 2 above a bad LexBFS triple, and vertex d a private neighbour of b with respect to c. Given a graph class G, a vertex ordering characterization (or VOC) of G is a total ordering on the vertices with specific properties, and ∀G, G ∈ G if and only if G admits a total ordering that satisfies said properties. VOCs have led to a number of efficient algorithms, and are often the basis of various graph recognition algorithms, see for instance [2,6,10,16,18]. We recall here some vertex ordering characterizations of the graph classes we consider.…”