Independent domination bondage number in graphs

TL;DR

The paper introduces the independent domination bondage number $b_{id}(G)$, the minimum number of edges whose removal changes the independent domination number $γ_i(G)$. It derives exact $b_{id}$ values for a variety of graph families (stars, complete bipartite graphs, friendship and book graphs, paths and cycles, complete graphs, cacti) and establishes general upper bounds in terms of minimum degree and $γ_i(G)$. It also studies how $b_{id}$ behaves under standard graph operations (join, lexicographic product $G[H]$, corona) and provides relationships that allow transferring known $b_{id}$ values across constructions. The results contribute to understanding the robustness of independent domination under edge deletions and have potential implications for network design and domination stability analyses.

Abstract

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent domination number of $G$, denoted by $γ_i(G)$, is the minimum size of all independent dominating sets of $G$. The independent domination bondage number of $G$ is the minimum number of edges whose removal changes the independent domination number of $G$. In this paper, we investigate properties of independent domination bondage number in graphs. In particular, we obtain several bounds and obtain the independent domination bondage number of some operations of two graphs.

Figures (5)

• Figure 1: The flowers $F_n$, $F_{4,n}$ and $F_{q,n}$, respectively.
• Figure 2: Book graph $B_n$.
• Figure 3: The chain triangular cactus $T_n$.
• Figure 4: para-chain square cactus graphs $Q_n$.
• Figure 5: ortho-chain square cactus $O_n$.

Theorems & Definitions (34)

• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Proposition 2.7
• proof
• Proposition 2.8
• proof
• Proposition 2.10
• proof