On the Kőnig‐Egerváry theorem for ‐paths

Abstract

The famous Kőnig‐Egerváry theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set G k of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order k equals the minimum order of a set of vertices intersecting all paths of order k. For k ∈ { 3 , 4 }, we give complete structural descriptions of the graphs in G k. Furthermore, for odd k, we give a complete structural description of the graphs in G k that contain no cycle of order less than k. For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order k and minimum sets of vertices intersecting all paths of order k.

Relating dissociation, independence, and matchings

“…For the hardness of deciding equality in (5), we describe the efficient construction of a graph H such that f is satisfiable if and only if diss(H) = α(H). For every clause C i = x ∨ y ∨ z in f , where x, y, and z are the three literals in C i , we introduce the six vertices x (i,1) , y (i,1) , z (i,1) , x (i,2) , y (i,2) , and z (i,2) in H that induce a subgraph H i that is a clique minus the three edges 1) x (i,2) , y (i,1) y (i,2) , and z (i,1) z (i,2) . Similarly as above, the vertices x (i,1) , y (i,1) , and z (i,1) in H i are associated with the three literals x, y, and z in C i .…”

“…As observed in several references, dissociations sets are the dual of so-called 3-path (vertex) covers, cf. also [1].…”