MATHEMATICA BOHEMICA, Vol. 126, No. 4, pp. 711-720 (2001)

# The forcing dimension of a graph

## Gary Chartrand, Ping Zhang

Gary Chartrand, Ping Zhang, Department of Math. and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA, e-mail: zhang@math-stat. wmich.edu

Abstract: For an ordered set $W=\{w_1, w_2, \cdots , w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1),d(v, w_2),\cdots , d(v, w_k)$), 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. A resolving set of minimum cardinality is a basis for $G$ and the number of vertices in a basis is its (metric) dimension $\dim (G)$. 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 $f_{G}(W, \dim )$ of $W$ in $G$ is the minimum cardinality of a forcing subset for $W$, while the forcing dimension $f(G, \dim )$ 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 $0 \leq a \leq b$ and $b \geq 1$, there exists a nontrivial connected graph $G$ with $f(G) = a$ and $\dim (G) = b$ if and only if $\{a, b\} \neq \{0, 1\}$.

Keywords: resolving set, basis, dimension, forcing dimension

Classification (MSC2000): 05C12

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