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The Steiner 3-diameter of a graph | | Yaping Mao
; | | Date: |
9 Sep 2015 | | Abstract: | The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian
and Zou in 1989, is a natural generalization of the concept of classical graph
distance. For a connected graph $G$ of order at least $2$ and $Ssubseteq
V(G)$, the emph{Steiner distance} $d(S)$ among the vertices of $S$ is the
minimum size among all connected subgraphs whose vertex sets contain $S$. Let
$n,k$ be two integers with $2leq kleq n$. Then the emph{Steiner
$k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=max
{d(S),|,Ssubseteq V(G), |S|=k, and vin S }$. Furthermore, the
emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=max {e_k(v),|, vin
V(G)}$. In 2011, Chartrand, Okamoto and Zhang showed that $k-1leq
sdiam_k(G)leq n-1$. In this paper, graphs with $sdiam_3(G)=2,3,n-1$ are
characterized, respectively. We also consider the Nordhaus-Gaddum-type results
for the parameter $sdiam_3(G)$. We determine sharp upper and lower bounds of
$sdiam_3(G)+sdiam_3(overline{G})$ and $sdiam_3(G)cdot sdiam_3(overline{G})$
for a graph $G$ of order $n$. Some graph classes attaining these bounds are
also given. | | Source: | arXiv, 1509.2801 | | Services: | Forum | Review | PDF | Favorites | |
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