212 research outputs found
Factor-Critical Property in 3-Dominating-Critical Graphs
A vertex subset of a graph is a dominating set if every vertex of
either belongs to or is adjacent to a vertex of . The cardinality of a
smallest dominating set is called the dominating number of and is denoted
by . A graph is said to be - vertex-critical if
, for every vertex in . Let be a 2-connected
-free 3-vertex-critical graph. For any vertex , we show
that has a perfect matching (except two graphs), which is a conjecture
posed by Ananchuen and Plummer.Comment: 8 page
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
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