k-Tuple_Total_Domination_in_Inflated_Graphs

TL;DR

This paper investigates the $k$-tuple total domination number in inflated graphs, providing bounds, characterizations, and calculations for various graph structures, extending understanding of domination parameters in complex graph transformations.

Contribution

It introduces bounds and characterizations for the $k$-tuple total domination number in inflated graphs when $k \\geq 2$, and analyzes its behavior in graphs with cut-edges or cut-vertices.

Findings

Bounds established: $n(G)k \\leq \, \gamma_{\\times k,t}(G_I) \\leq \, n(G)(k+1)-1$

Characterizations for graphs with minimal or near-minimal $k$-tuple total domination number in inflated graphs

Formulas for the number in graphs with specific structural features like cut-edges or cut-vertices.

Abstract

The inflated graph $G_{I}$ of a graph $G$ with $n(G)$ vertices is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $u\in X_{i}$, $v\in X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. For integer $k\geq 1$, the $k$-tuple total domination number $\gamma_{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$, which is a set of vertices in $G$ such that every vertex of $G$ is adjacent to at least $k$ vertices in it. For existing this number, must the minimum degree of $G$ is at least $k$. Here, we study the $k$-tuple total domination number in inflated graphs when $k\geq 2$. First we prove that $n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1$, and then we characterize graphs $G$ that the $k$-tuple total domination number number of $G_I$ is $n(G)k$ or $n(G)k+1$. Then we find bounds for this number in the inflated graph $G_I$, when $G$ has a cut-edge $e$ or cut-vertex $v$, in terms on the $k$-tuple total domination number of the inflated graphs of the components of $G-e$ or $v$-components of $G-v$, respectively. Finally, we calculate this number in the inflated graphs that have obtained by some of the known graphs.