NDLI: Total Domination Edge Critical Graphs with Total Domination Number Three and Many Dominating Pairs

Content Provider	Springer Nature Link
Author	Haynes, Teresa W. Balbuena, Cami Hansberg, Adriana Henning, Michael A.
Copyright Year	2014
Abstract	A graph $$G$$ is diameter-2-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-2-critical graph $$G$$ of order $$n$$ is at most $$\lfloor n^2/4 \rfloor $$ and that the extremal graphs are the complete bipartite graphs $$K_{{\lfloor n/2 \rfloor },{\lceil n/2 \rceil }}$$ . A graph is $$3_t$$ -edge-critical, abbreviated $$3_tEC$$ , if its total domination number is 3 and the addition of any edge decreases the total domination number. It is known that proving the Murty–Simon Conjecture is equivalent to proving that the number of edges in a $$3_tEC$$ graph of order $$n$$ is greater than $$\lceil n(n-2)/4 \rceil $$ . We study a family $$\mathcal{F}$$ of $$3_tEC$$ graphs of diameter 2 for which every pair of nonadjacent vertices dominates the graph. We show that the graphs in $$\mathcal{F}$$ are precisely the bull-free $$3_tEC$$ graphs and that the number of edges in such graphs is at least $$\lfloor (n^2 - 4)/4 \rfloor $$ , proving the conjecture for this family. We characterize the extremal graphs, and conjecture that this improved bound is in fact a lower bound for all $$3_tEC$$ graphs of diameter 2. Finally we slightly relax the requirement in the definition of $$\mathcal{F}$$ —instead of requiring that all pairs of nonadjacent vertices dominate to requiring that only most of these pairs dominate—and prove the Murty–Simon equivalent conjecture for these $$3_tEC$$ graphs.
Ending Page	1176
Page Count	14
Starting Page	1163
File Format	PDF
ISSN	09110119
e-ISSN	14355914
Journal	Graphs and Combinatorics
Issue Number	5
Volume Number	31
Language	English
Publisher	Springer Japan
Publisher Date	2014-10-04
Publisher Place	Tokyo
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Domination Bull-free Diameter-2-critical Total domination edge critical Dominating sets, independent sets, cliques Engineering Design Hypergraphs Combinatorics Diameter critical
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science

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