Given an undirected graph G = (V , E) with matching number ν(G), we define d-blockers as subsets of edges B such that ν((V , E \ B)) ≤ ν(G) − d. We define d-transversals T as subsets of edges such that every maximum matching M has |M ∩ T | ≥ d. We explore connections between d-blockers and d-transversals. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, chains and cycles and we construct minimum d-transversals and d-blockers in these special graphs. We also study the complexity status of finding minimum transversals and blockers in arbitrary graphs.

Starting from the basic problem of reconstructing a 2-dimensional image given by its projections on two axes, one associates a model of edge coloring in a complete bipartite graph. The complexity of the case with k = 3 colors is open. Variations and special cases are considered for the case k = 3 colors where the graph corresponding to the union of some color classes (for instance colors 1 and 2) has a given structure (tree, vertexdisjoint chains, 2-factor, etc.). We also study special cases corresponding to the search of 2 edge-disjoint chains or cycles going through specified vertices. A variation where the graph is oriented is also presented.In addition we explore similar problems for the case where the underlying graph is a complete graph (instead of a complete bipartite graph).

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