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 Tr. Inst. Mat., 2010, Volume 18, Number 2, Pages 79–86 (Mi timb19)

The profile of the corona $G\wedge H$, where $G$ is a Halin graph, whose tree is a caterpillar

V. V. Lepin, S. A. Tsikhan

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: Let $G=(V,E)$ be a graph on $n$ vertices. A 1-1 mapping $f\colon V\to\{1,2,…,n\}$ is called a linear arrangement of $G$. Given a graph $G$, the profile problem is to find the profile of
$$G:p(G)=\min_f\sum_{v\in V}\max_{u\in N[v]}(f(v)-f(u)),$$
where $N[v]=\{v\}\cup\{u\in V:\{v,u\}\in E\}$. A Halin graph $H=T\cup C$ is obtained by embedding a tree $T$ having no degree two nodes in the plane, and then adding a cycle $C$ to join the leaves of $T$ in such a way that the resulting graph is planar. The corona of graphs $G_1$ and $G_2$, on $n_1$ and $n_2$ vertices, respectively, is denoted by $G_1\wedge G_2$ and contains one copy of $G_1$ and $n_1$ copies of $G_2$. Each distinct vertex of $G_1$ is joined to every vertex of the corresponding copy of $G_2$. This paper shows that, if $G$ is a Halin graph such that the tree $T$ is a caterpillar then $p(G)=3(n-2)$ and $p(G\wedge H)=3(n-2)+np(H)+(3n-6)m$, where $n=|V(G)|$, $m=|V(H)|$.

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Bibliographic databases:
UDC: 519.1

Citation: V. V. Lepin, S. A. Tsikhan, “The profile of the corona $G\wedge H$, where $G$ is a Halin graph, whose tree is a caterpillar”, Tr. Inst. Mat., 18:2 (2010), 79–86

Citation in format AMSBIB
\Bibitem{LepTsi10} \by V.~V.~Lepin, S.~A.~Tsikhan \paper The profile of the corona $G\wedge H$, where $G$ is a Halin graph, whose tree is a caterpillar \jour Tr. Inst. Mat. \yr 2010 \vol 18 \issue 2 \pages 79--86 \mathnet{http://mi.mathnet.ru/timb19} \zmath{https://zbmath.org/?q=an:05863492}