12,269 research outputs found
On Murty-Simon Conjecture II
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 vertices is at most
and the extremal graph is the complete
bipartite graph .
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 , where ; 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
A vertex subset of a graph is a double dominating set of if
for each vertex of , where is the set of the
vertex and vertices adjacent to . The double domination number of ,
denoted by , is the cardinality of a smallest double
dominating set of . A graph is said to be double domination edge
critical if for any edge . A double domination edge critical graph with is called --critical. A graph is
-factor-critical if has a perfect matching for each set of
vertices in . In this paper we show that is 3-factor-critical if is
a 3-connected claw-free --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
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
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