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Characterization of Cubic Graphs G with ir t (G) = Ir t (G) = 2

Changiz Eslahchi;  Shahab Haghi;  Nader Jafari — 2014

Discussiones Mathematicae Graph Theory

A subset S of vertices in a graph G is called a total irredundant set if, for each vertex v in G, v or one of its neighbors has no neighbor in S −{v}. The total irredundance number, ir(G), is the minimum cardinality of a maximal total irredundant set of G, while the upper total irredundance number, IR(G), is the maximum cardinality of a such set. In this paper we characterize all cubic graphs G with irt(G) = IRt(G) = 2

On Double-Star Decomposition of Graphs

Saieed Akbari;  Shahab Haghi;  Hamidreza Maimani;  Abbas Seify — 2017

Discussiones Mathematicae Graph Theory

A tree containing exactly two non-pendant vertices is called a double-star. A double-star with degree sequence (k1 + 1, k2 + 1, 1, . . . , 1) is denoted by Sk1,k2. We study the edge-decomposition of graphs into double-stars. It was proved that every double-star of size k decomposes every 2k-regular graph. In this paper, we extend this result by showing that every graph in which every vertex has degree 2k + 1 or 2k + 2 and containing a 2-factor is decomposed into Sk1,k2 and Sk1−1,k2, for all positive...

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