On the path sequence of a graph

A subset S of vertices of a graph G = (V;E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by k(G) the minimum cardinality of a k-path vertex cover in G and form a sequence (G) = ( 1(G); 2(G); : : : ; jV j(G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in (G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given

On the Path Sequence of a Graph

A subset S of vertices of a graph G = (V,E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by Ψk (G) the minimum cardinality of a k-path vertex cover in G and form a sequence Ψ (G) = (Ψ1 (G), Ψ2 (G), . . . , Ψ|V| (G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in Ψ(G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given

Uniformly dissociated graphs

Creative Commons Attribution 3.0 International LicenseA set D of vertices in a graph G is called a dissociation set if every vertex in D has at most one neighbor in D. We call a graph G uniformly dissociated if all maximal dissociation sets are of the same cardinality. Characterizations of uniformly dissociated graphs with small cardinalities of dissociation sets are proven; in particular, the graphs in which all maximal dissociation sets are of cardinality 2 are the complete graphs on at least two vertices from which possibly a matching is removed, while the graphs in which all maximal dissociation sets are of cardinality 3 are the complements of the K4-free geodetic graphs with diameter 2. A general construction by which any graph can be embedded as an induced sub graph of a uniformly dissociated graph is also presented. In the main result we characterize uniformly dissociated graphs with girth at least 7 to be either isomorphic to C7, or obtainable from an arbitrary graph H with girth at least 7 by identifying each vertex of H with a leaf of a copy of P3

On the k-path vertex cover of some graph products

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ▫$psi_k$▫(G) the minimum cardinality of a k-path vertex cover in G. In this paper, improved lower and upper bounds for ▫$psi_k$▫ of the Cartesian and the strong product of paths are derived. It is shown that for ▫$psi_3$▫ those bounds are tight. For the lexicographic product bounds are presented for ▫$psi_k$▫, moreover ▫$psi_2$▫ and ▫$psi_3$▫ are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given