Displaying similar documents to “Secure sets and their expansion in cubic graphs”

Graphs with disjoint dominating and paired-dominating sets

Justin Southey, Michael Henning (2010)

Open Mathematics

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A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned...

The ramsey number for theta graph versus a clique of order three and four

M.S.A. Bataineh, M.M.M. Jaradat, M.S. Bateeha (2014)

Discussiones Mathematicae Graph Theory

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For any two graphs F1 and F2, the graph Ramsey number r(F1, F2) is the smallest positive integer N with the property that every graph on at least N vertices contains F1 or its complement contains F2 as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. We determine r(θn,Km) for m = 2, 3, 4 and n > m. More specifically, we establish that r(θn,Km) = (n − 1)(m − 1) + 1 for m = 3, 4 and n > m