Metric dimension of maximal outerplanar graphs

Abstract

In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if $β(G)$ is the metric dimension of a maximal outerplanar graph $G$ of order $n$, we prove that $2\le β(G) \le \lceil \frac{2n}{5}\rceil$ and that the bounds are tight. We also provide linear algorithms to decide whether the metric dimension of $G$ is 2 and to build a resolving set of size $\lceil \frac{2n}{5}\rceil$ for $G$. Moreover, we characterize the maximal outerplanar graphs with metric dimension 2.

Figures (16)

• Figure 1: Part (a): Given a 2-tree $G$ with metric dimension 2, a minimal induced 2-connected subgraph containing the basis $\{a_1,a_k\}$, as claimed in BDJO17. Part (b): A 2-tree with metric basis $\{a_1,a_k\}$ whose minimal induced 2-connected subgraph containing $a_1$ and $a_k$ is different from the claimed subgraph in BDJO17.
• Figure 2: Left: A graph $G$ with metric basis $S=\{1,3\}$. The metric coordinates of the vertices are: $r(1|S) = (0,2), r(2|S) = (1,1), r(3|S) = (2,0), r(4|S) = (3,1), r(5|S) = (2,1), r(6|S) = (3,2), r(7|S) = (3,3), r(8|S) = (2,3), r(9|S) = (1,3), r(10|S) = (1,2)$ and $r(11|S) = (2,2)$. Right: The representation $G^*$ of $G$ as a subgraph of $P_n\boxtimes P_n$ with respect to $S$. Vertex $v$ in $G$ is mapped to vertex $v^*$ in $G^*$ such that the cartesian coordinates of $v^*$ are the metric coordinates of $v$.
• Figure 3: Left: Illustrating Proposition \ref{['prop.reprDim2']}. The shortest path from $(0,d)$ to $(d,0)$ and the set $A_d$, which is in the shaded region. Right: Examples of horizontal and vertical MOP zigzags.
• Figure 4: An example of a MOP graph $G$ with metric dimension 2. If the vertices of a basis are at distance $d$, then $G$ can be represented as a subgraph $G^*$ of the strong product $P_n\boxtimes P_n$ such that all vertices of $G^*$ belong to the shaded region. Vertices described in Theorem \ref{['thm:caractMOPs']} (1), (2), (3) and (4) are added in $(a)$, $(b)$, $(c)$ and $(d)$, respectively. Observe that all vertices of $G^*$ belong to the unbounded face.
• Figure 5: A $(0,d)-(i,j)$ path of length $i$ when $i-j$ and $d$ have distinct parity (left); when $i-j$ and $d$ have the same parity and $i-j\not=-d$ (center) and when $i-j=-d$ (right).

Theorems & Definitions (17)

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