# The forcing dimension of a graph

Mathematica Bohemica (2001)

• Volume: 126, Issue: 4, page 711-720
• ISSN: 0862-7959

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## Abstract

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For an ordered set $W=\left\{{w}_{1},{w}_{2},\cdots ,{w}_{k}\right\}$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r\left(v|W\right)$ = ($d\left(v,{w}_{1}\right),d\left(v,{w}_{2}\right),\cdots ,d\left(v,{w}_{k}\right)$), where $d\left(x,y\right)$ 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. A resolving set of minimum cardinality is a basis for $G$ and the number of vertices in a basis is its (metric) dimension $dim\left(G\right)$. For a basis $W$ of $G$, a subset $S$ of $W$ is called a forcing subset of $W$ if $W$ is the unique basis containing $S$. The forcing number ${f}_{G}\left(W,dim\right)$ of $W$ in $G$ is the minimum cardinality of a forcing subset for $W$, while the forcing dimension $f\left(G,dim\right)$ of $G$ is the smallest forcing number among all bases of $G$. The forcing dimensions of some well-known graphs are determined. It is shown that for all integers $a,b$ with $0\le a\le b$ and $b\ge 1$, there exists a nontrivial connected graph $G$ with $f\left(G\right)=a$ and $dim\left(G\right)=b$ if and only if $\left\{a,b\right\}\ne \left\{0,1\right\}$.

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