8 research outputs found

    General dd-position sets

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    The general dd-position number gpd(G){\rm gp}_d(G) of a graph GG is the cardinality of a largest set SS for which no three distinct vertices from SS lie on a common geodesic of length at most dd. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of gpd(G){\rm gp}_d(G) with respect to the suitable values of dd. We show that the decision problem concerning finding gpd(G){\rm gp}_d(G) is NP-complete for any value of dd. The value of gpd(G){\rm gp}_d(G) when GG is a path or a cycle is computed and a structural characterization of general dd-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that gpd(G){\rm gp}_d(G) is infinite whenever GG is an infinite graph and dd is a finite integer.Comment: 16 page

    On the path sequence of a graph

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    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

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    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

    On computing the minimum 3-path vertex cover and dissociation number of graphs

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    The dissociation number of a graph G is the number of vertices in a maximum size induced subgraph of G with vertex degree at most 1. A k-path vertex cover of a graph G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S. The minimum 3-path vertex cover is a dual problem to the dissociation number. For this problem we present an exact algorithm with a running time of O ∗ (1.5171 n) on a graph with n vertices. We also provide a polynomial time randomized approximation algorithm with an for the minimum 3-path vertex cover. expected approximation ratio of 23/11 for th
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