The forcing dimension of a graph

Gary Chartrand; Ping Zhang

Displaying similar documents to “The forcing dimension of a graph”

Resolving domination in graphs

Robert C. Brigham, Gary Chartrand, Ronald D. Dutton, Ping Zhang (2003)

Mathematica Bohemica

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For an ordered set $W = {w_{1}, w_{2}, \dots, w_{k}}$ of vertices and a vertex $v$ in a connected graph $G$ , the (metric) representation of $v$ with respect to $W$ is the $k$ -vector $r (v | W) = (d (v, w_{1}), d (v, w_{2}), \dots, d (v, w_{k}))$ , where $d (x, y)$ represents the distance between the vertices $x$ and $y$ . The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$ . A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $dim G$ . A set $S$ of vertices in $G$ is a dominating...

Connected resolvability of graphs

Varaporn Saenpholphat, Ping Zhang (2003)

Czechoslovak Mathematical Journal

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For an ordered set $W = {w_{1}, w_{2}, \dots, w_{k}}$ of vertices and a vertex $v$ in a connected graph $G$ , the representation of $v$ with respect to $W$ is the $k$ -vector $r (v | W)$ = ( $d (v, w_{1})$ , $d (v, w_{2}), \dots, d (v, w_{k}))$ , where $d (x, y)$ represents the distance between the vertices $x$ and $y$ . The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$ . A resolving set for $G$ containing a minimum number of vertices is a basis for $G$ . The dimension $dim (G)$ is the number of vertices in a basis for $G$ . A resolving set $W$ of $G$ is connected...

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

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The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_{f}$ of all finite undirected graphs. If G is a graph from $C_{f}$ , then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.