### AIMS Mathematics

2021, Issue 2: 1140-1152. doi: 10.3934/math.2021069
Research article

# Fault-tolerant edge metric dimension of certain families of graphs

• Received: 23 July 2020 Accepted: 16 October 2020 Published: 11 November 2020
• MSC : 68R01, 68R05, 68R10

• Let $W_E = \{w_1, w_2, \ldots, w_k\}$ be an ordered set of vertices of graph $G$ and let $e$ be an edge of $G$. Suppose $d(x, e)$ denotes distance between edge $e$ and vertex $x$ of $G$, defined as $d(e, x) = d(x, e) = \min \{d(x, a), d(x, b)\}$, where $e = ab$. A vertex $x$ distinguishes two edges $e_1$ and $e_2$, if $d(e_1, x)\neq d(e_2, x)$. The representation $r(e\mid W_E)$ of $e$ with respect to $W_E$ is the k-tuple $(d(e, w_1), d(e, w_2), \ldots, d(e, w_k))$. If distinct edges of $G$ have distinct representation with respect to $W_E$, then $W_E$ is called edge metric generator for $G$. An edge metric generator of minimum cardinality is an edge metric basis for $G$, and its cardinality is called edge metric dimension of $G$, denoted by $(G)$. In this paper, we initiate the study of fault-tolerant edge metric dimension. Let $\acute{W}_E$ be edge metric generator of graph $G$, then $\acute{W}_E$ is called fault-tolerant edge metric generator of $G$ if $\acute{W}_E \setminus \{v \}$ is also an edge metric generator of graph $G$ for every $v \in \acute{W}_E$. A fault-tolerant edge metric generator of minimum cardinality is a fault-tolerant edge metric basis for graph $G$, and its cardinality is called fault-tolerant edge metric dimension of $G$. We also computed the fault-tolerant edge metric dimension of path, cycle, complete graph, cycle with chord graph, tadpole graph and kayak paddle graph.

Citation: Xiaogang Liu, Muhammad Ahsan, Zohaib Zahid, Shuili Ren. Fault-tolerant edge metric dimension of certain families of graphs[J]. AIMS Mathematics, 2021, 6(2): 1140-1152. doi: 10.3934/math.2021069

### Related Papers:

• Let $W_E = \{w_1, w_2, \ldots, w_k\}$ be an ordered set of vertices of graph $G$ and let $e$ be an edge of $G$. Suppose $d(x, e)$ denotes distance between edge $e$ and vertex $x$ of $G$, defined as $d(e, x) = d(x, e) = \min \{d(x, a), d(x, b)\}$, where $e = ab$. A vertex $x$ distinguishes two edges $e_1$ and $e_2$, if $d(e_1, x)\neq d(e_2, x)$. The representation $r(e\mid W_E)$ of $e$ with respect to $W_E$ is the k-tuple $(d(e, w_1), d(e, w_2), \ldots, d(e, w_k))$. If distinct edges of $G$ have distinct representation with respect to $W_E$, then $W_E$ is called edge metric generator for $G$. An edge metric generator of minimum cardinality is an edge metric basis for $G$, and its cardinality is called edge metric dimension of $G$, denoted by $(G)$. In this paper, we initiate the study of fault-tolerant edge metric dimension. Let $\acute{W}_E$ be edge metric generator of graph $G$, then $\acute{W}_E$ is called fault-tolerant edge metric generator of $G$ if $\acute{W}_E \setminus \{v \}$ is also an edge metric generator of graph $G$ for every $v \in \acute{W}_E$. A fault-tolerant edge metric generator of minimum cardinality is a fault-tolerant edge metric basis for graph $G$, and its cardinality is called fault-tolerant edge metric dimension of $G$. We also computed the fault-tolerant edge metric dimension of path, cycle, complete graph, cycle with chord graph, tadpole graph and kayak paddle graph.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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