Signpost systems and spanning trees of graphs

Title:

Signpost systems and spanning trees of graphs

Creator:

Nebeský, Ladislav

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:b001ba78-8f1c-43ae-a4f4-c14f415b2a6f
uuid:b001ba78-8f1c-43ae-a4f4-c14f415b2a6f

Subject:

signpost system, path, connected graph, tree, and spanning tree

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

885-893

Source:

Czechoslovak Mathematical Journal | 2006 Volume:56 | Number:3

Harvested from:

CDK

Metadata only:

false