In: Graphs and Combinatorics, 2007, vol. 23, no. 1, p. 47-60
We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R| = n + 1 such that perfect matchings with k red edges exist for all k, 0 ≤ k ≤ n. Given two integers p...
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In: Discrete Mathematics, 2012, vol. 312, no. 10, p. 1838-1843
In threshold graphs one may find weights for the vertices and a threshold value t such that for any subset S of vertices, the sum of the weights is at most the threshold t if and only if the set S is a stable (independent) set. In this note we ask a similar question about vertex colorings: given an integer p, when is it possible to find weights (in general depending on p) for the vertices and a...
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In: Discrete Mathematics and Theoretical Computer Science, 2010, vol. 12, no. 5, p. 1-24
The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some...
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In: Discrete Mathematics, 2010, vol. 310, p. 132-146
Given an undirected graph G=(V,E) with matching number \nu(G), a d-blocker is a subset of edges B such that \nu(/V,E\B))= d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the...
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In: Journal of Discrete Algorithms, 2012, vol. 17, p. 95-102
Let G = (V , E) be a graph in which every vertex v ∈ V has a weight w(v)>=0 and a cost c(v) >=0. Let SG be the family of all maximum-weight stable sets in G. For any integer d 0, a minimum d-transversal in the graph G with respect to SG is a subset of vertices T ⊆ V of minimum total cost such that |T ∩ S| d for every S ∈ SG. In this paper, we present a polynomial-time algorithm to...
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