Signpost systems and spanning trees of graphs
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 885-893
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
By a ternary system we mean an ordered pair $(W, R)$, where $W$ is a finite nonempty set and $R \subseteq W \times W \times W$. By a signpost system we mean a ternary system $(W, R)$ satisfying the following conditions for all $x, y, z \in W$: if $(x, y, z) \in R$, then $(y, x, x) \in R$ and $(y, x, z) \notin R$; if $x \ne y$, then there exists $t \in W$ such that $(x, t, y) \in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G, T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G, T)$ is a ct-pair, then by the guide to $(G,T)$ we mean the ternary system $(W, R)$, where $W = V(G)$ and the following condition holds for all $u, v, w \in W$: $(u, v, w) \in R$ if and only if $uv \in E(G)$ and $v$ belongs to the $u-w$ path in $T$. By Proposition 1, the guide to a ct-pair is a signpost system. We say that a signpost system is tree-controlled if it satisfies a certain set of four axioms (these axioms could be formulated in a language of the first-order logic). Consider the mapping $\phi $ from the class of all ct-pairs into the class of all signpost systems such that $\phi ((G, T))$ is the guide to $(G, T)$ for every ct-pair $(G, T)$. It is proved in this paper that $\phi $ is a bijective mapping from the class of all ct-pairs onto the class of all tree-controlled signpost systems.
By a ternary system we mean an ordered pair $(W, R)$, where $W$ is a finite nonempty set and $R \subseteq W \times W \times W$. By a signpost system we mean a ternary system $(W, R)$ satisfying the following conditions for all $x, y, z \in W$: if $(x, y, z) \in R$, then $(y, x, x) \in R$ and $(y, x, z) \notin R$; if $x \ne y$, then there exists $t \in W$ such that $(x, t, y) \in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G, T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G, T)$ is a ct-pair, then by the guide to $(G,T)$ we mean the ternary system $(W, R)$, where $W = V(G)$ and the following condition holds for all $u, v, w \in W$: $(u, v, w) \in R$ if and only if $uv \in E(G)$ and $v$ belongs to the $u-w$ path in $T$. By Proposition 1, the guide to a ct-pair is a signpost system. We say that a signpost system is tree-controlled if it satisfies a certain set of four axioms (these axioms could be formulated in a language of the first-order logic). Consider the mapping $\phi $ from the class of all ct-pairs into the class of all signpost systems such that $\phi ((G, T))$ is the guide to $(G, T)$ for every ct-pair $(G, T)$. It is proved in this paper that $\phi $ is a bijective mapping from the class of all ct-pairs onto the class of all tree-controlled signpost systems.
Classification :
05C05, 05C12, 05C38, 05C99, 90B10
Keywords: signpost system; path; connected graph; tree; spanning tree
Keywords: signpost system; path; connected graph; tree; spanning tree
@article{CMJ_2006_56_3_a6,
author = {Nebesk\'y, Ladislav},
title = {Signpost systems and spanning trees of graphs},
journal = {Czechoslovak Mathematical Journal},
pages = {885--893},
year = {2006},
volume = {56},
number = {3},
mrnumber = {2261660},
zbl = {1164.05392},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a6/}
}
Nebeský, Ladislav. Signpost systems and spanning trees of graphs. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 885-893. http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a6/
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