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A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G − v is less than the total domination number of G. These graphs we call γ t-critical. If such a graph G has total domination number k, we call it k-γ t-critical. We characterize the(More)
A defensive alliance in a graph G = (V, E) is a set of vertices S ⊆ V satisfying the condition that for every vertex v ∈ S, the number of neighbors v has in S plus one (counting v) is at least as large as the number of neighbors it has in V − S. Because of such an alliance, the vertices in S, agreeing to mutually support each other, have the strength of(More)
In this paper we study tight lower bounds on the size of a maximum matching in a regular graph. For k ≥ 3, let G be a connected k-regular graph of order n and let α (G) be the size of a maximum matching in G. We show that if k is even, then α (G) ≥ min k 2 +4 k 2 +k+2 × n 2 , n−1 2 , while if k is odd, then α (G) ≥ (k 3 −k 2 −2) n−2k+2 2(k 3 −3k). We show(More)