3,656 research outputs found

    On Murty-Simon Conjecture II

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    A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on nn vertices is at most ⌊n24⌋\lfloor \frac{n^{2}}{4} \rfloor and the extremal graph is the complete bipartite graph K⌊n2⌋,⌈n2⌉K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}. In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al. is not the original conjecture, indeed, it is only for the diameter two edge-critical graphs of even order. In this paper, we completely prove the Murty-Simon Conjecture for the graphs whose complements have vertex connectivity ℓ\ell, where ℓ=1,2,3\ell = 1, 2, 3; and for the graphs whose complements have an independent vertex cut of cardinality at least three.Comment: 9 pages, submitted for publication on May 10, 201

    Locating-total dominating sets in twin-free graphs: a conjecture

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    A total dominating set of a graph GG is a set DD of vertices of GG such that every vertex of GG has a neighbor in DD. A locating-total dominating set of GG is a total dominating set DD of GG with the additional property that every two distinct vertices outside DD have distinct neighbors in DD; that is, for distinct vertices uu and vv outside DD, N(u)∩D≠N(v)∩DN(u) \cap D \ne N(v) \cap D where N(u)N(u) denotes the open neighborhood of uu. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of GG, denoted LT(G)LT(G), is the minimum cardinality of a locating-total dominating set in GG. It is well-known that every connected graph of order n≥3n \geq 3 has a total dominating set of size at most 23n\frac{2}{3}n. We conjecture that if GG is a twin-free graph of order nn with no isolated vertex, then LT(G)≤23nLT(G) \leq \frac{2}{3}n. We prove the conjecture for graphs without 44-cycles as a subgraph. We also prove that if GG is a twin-free graph of order nn, then LT(G)≤34nLT(G) \le \frac{3}{4}n.Comment: 18 pages, 1 figur

    Total domination stable graphs upon edge addition

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    AbstractA set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs
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