8 research outputs found
General -position sets
The general -position number of a graph is the
cardinality of a largest set for which no three distinct vertices from
lie on a common geodesic of length at most . This new graph parameter
generalizes the well studied general position number. We first give some
results concerning the monotonic behavior of with respect to
the suitable values of . We show that the decision problem concerning
finding is NP-complete for any value of . The value of when is a path or a cycle is computed and a structural
characterization of general -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 is
infinite whenever is an infinite graph and is a finite integer.Comment: 16 page
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
On computing the minimum 3-path vertex cover and dissociation number of graphs
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