The line graph of a graph , denoted by , has as its vertex set, where two vertices in are adjacent if and only if the corresponding edges in have a vertex in common. For a graph , define . Let be a 2-connected claw-free simple graph of order with . We show that, if and is sufficiently large, then either is traceable or the Ryjáček’s closure , where is an essentially -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10 which have...
For a graph G = (V,E), a set S ⊆ V(G) is a total dominating set if it is dominating and both ⟨S⟩ has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V(G) is a total restrained dominating set if it is total dominating and ⟨V(G)-S⟩ has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained...
Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by , is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds...
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