Liliek Susilowati | Searcharxiv

arXiv2026

The Diametral Metric Dimension of Generalized Corona Graph

This study investigates the diametral metric dimension of generalized corona graphs, where the main graph is connected graph and the branch graphs form a sequence of connected graphs. The concept of diametral metric dimension is an extension of the metric dimension concept, requiring the resolving set to contain all diametral vertices. To determine the diametral metric dimension of a generalized corona graph, one must first identify the distance of each vertex in the graph and the graph's diametral set. Subsequently, a resolving set containing the diametral set is determined. The results show that the diametral metric dimension of the generalized corona graph depends on the diameter of the main graph and the metric dimensions of the graphs in the sequence. These concepts provide theoretical insights into resolving structures in the planning infrastructure and motivate further studies on other graph families and graph operations.

math.CO↗

arXiv2022

A study of a combination of distance domination and resolvability in graphs

For $k \geq 1$, in a graph $G=(V,E)$, a set of vertices $D$ is a distance $k$-dominating set of $G$, if any vertex in $V\setminus D$ is at distance at most $k$ from some vertex in $D$. The minimum cardinality of a distance $k$-dominating set of $G$ is the distance $k$-domination number, denoted by $γ_k(G)$. An ordered set of vertices $W=\{w_1,w_2,\ldots,w_r\}$ is a resolving set of $G$, if for any two distinct vertices $x$ and $y$ in $V\setminus W$, there exists $1\leq i\leq r$, such that $d_G(x,w_i)\neq d_G(y,w_i)$. The minimum cardinality of a resolving set of $G$ is the metric dimension of the graph $G$, denoted by $dim(G)$. In this paper, we introduce the distance $k$-resolving dominating set, which is a subset of $V$ that is both a distance $k$-dominating set and a resolving set of $G$. The minimum cardinality of a distance $k$-resolving dominating set of $G$ is called the distance $k$-resolving domination number and is denoted by $γ^r_k(G)$. We give several bounds for $γ^r_k(G)$ some in terms of the metric dimension $dim(G)$ and the distance $k$-domination number $γ_k(G)$. We determine $γ^r_k(G)$ when $G$ is a path or a cycle. Afterwards, we characterize the connected graphs of order $n$ having $γ^r_k(G)$ equal to $1$, $n-2$, and $n-1$, for $k\geq 2$. Then, we construct graphs realizing all the possible triples $(dim(G),γ_k(G),γ^r_k (G))$, for all $k\geq 2$. Later, we determine the maximum order of a graph $G$ having distance $k$-resolving domination number $γ^r_k(G)=γ^r_k\geq 1$, we provide graphs achieving this maximum order for any positive integers $k$ and $γ^r_k$. Finally, we establish Nordhaus-Gaddum bounds for $γ^r_k(G)$, for $k\geq 2$.

math.CO↗