Toughness of the corona of two graphs

The toughness of a non-complete graph G = (V , E) is defined as τ (G) = min{|S|/ω(G − S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G − S) denotes the number of components of the resultant graph G − S by deletion of S. The corona of two graphs G and H , written as G ◦ H , is the graph obtained by taking one copy of G and |V (G)| copies of H , and then joining the ith vertex of G to every vertex in the ith copy of H . In this paper, we investigate the toughness of this kind of graphs and obtain the exact value for the corona of two graphs belonging to some families as paths, cycles, stars, wheels or complete graphs.Ministerio de Educación y Ciencia MTM2008-06620-C03-02Generalitat de Cataluña 1298 SGR2009Junta de Andalucía P06-FQM-0164

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Neighbor Isolated Tenacity of Graphs

The tenacity of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the neighbor isolated version of this parameter. The neighbor isolated tenacity of a noncomplete connected graph G is defined to be NIT(G) = min {|X|+ c(G/X) / i(G/X), i(G/X) ≥ 1} where the minimum is taken over all X, the cut strategy of G , i(G/X)is the number of components which are isolated vertices of G/X and c(G/X) is the maximum order of the components of G/X. Next, the relations between neighbor isolated tenacity and other parameters are determined and the neighbor isolated tenacity of some special graphs are obtained. Moreover, some results about the neighbor isolated tenacity of graphs obtained by graph operations are given