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In this paper, we survey some new results in four areas of domination in graphs, namely:
(1) the toughness and matching structure of graphs having domination number 3 and which are "critical" in the sense that if one adds any missing edge, the domination number falls to 2;
(2) the matching structure of graphs having domination number 3 and which are "critical" in the sense that if one deletes any vertex, the domination number falls to 2;
(3) upper bounds...
François Jaeger conjectured in 1974 that every cyclically 4-connected cubic graph is dual hamiltonian, that is to say the vertices of can be partitioned into two subsets such that each subset induces a tree in . We shall make several remarks on this conjecture.
A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work we analyze the structure of strong k-transitive digraphs having a cycle of length at least k. We show...
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