12,269 research outputs found

    On Murty-Simon Conjecture II

    Full text link
    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 ⌊n24βŒ‹\lfloor \frac{n^{2}}{4} \rfloor and the extremal graph is the complete bipartite graph K⌊n2βŒ‹,⌈n2βŒ‰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 β„“\ell, where β„“=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

    3-Factor-criticality in double domination edge critical graphs

    Full text link
    A vertex subset SS of a graph GG is a double dominating set of GG if ∣N[v]∩S∣β‰₯2|N[v]\cap S|\geq 2 for each vertex vv of GG, where N[v]N[v] is the set of the vertex vv and vertices adjacent to vv. The double domination number of GG, denoted by Ξ³Γ—2(G)\gamma_{\times 2}(G), is the cardinality of a smallest double dominating set of GG. A graph GG is said to be double domination edge critical if Ξ³Γ—2(G+e)<Ξ³Γ—2(G)\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G) for any edge eβˆ‰Ee \notin E. A double domination edge critical graph GG with Ξ³Γ—2(G)=k\gamma_{\times 2}(G)=k is called kk-Ξ³Γ—2(G)\gamma_{\times 2}(G)-critical. A graph GG is rr-factor-critical if Gβˆ’SG-S has a perfect matching for each set SS of rr vertices in GG. In this paper we show that GG is 3-factor-critical if GG is a 3-connected claw-free 44-Ξ³Γ—2(G)\gamma_{\times 2}(G)-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page

    Protecting a Graph with Mobile Guards

    Full text link
    Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve
    • …
    corecore