Partial domination of middle graphs

TL;DR

The work studies partial domination on middle graphs by proving that the isolation number of $\mathrm{Mid}(G)$ equals the minimal maximal matching size in $G$, i.e. $\iota(\mathrm{Mid}(G))=\nu'(G)$, via the intermediate parameter $\tau(G)$. It provides exact values for paths and cycles, tight general bounds, and an extensive treatment of trees, including extremal and structural characterizations. The results connect partial domination in transformation graphs to classical maximal matching theory, yielding precise extremal results for complete, complete bipartite, and tree families and guiding future exploration of other transformation graphs. Overall, the paper establishes a tight and versatile bridge between isolation-based domination in transformed graphs and maximal matching parameters in the original graph, with both theoretical and potential algorithmic implications.

Abstract

For any graph $G=(V,E)$, a subset $S\subseteq V$ is called {\it an isolating set} of $G$ if $V\setminus N_G[S]$ is an independent set of $G$, where $N_G[S]=S\cup N_G(S)$, and {\it the isolation number} of $G$, denoted by $ι(G)$, is the size of a smallest isolating set of $G$. In this article, we show that the isolation number of the middle graph of $G$ is equal to the size of a smallest maximal matching of $G$.

Figures (6)

• Figure 1: $G$ and its middle graph ${Mid}(G)$
• Figure 2: $E_0\subseteq E(G)$
• Figure 3: $\iota({Mid}(T_i))=k= \left \lfloor {\frac{|V(T_i)|-1}{2}}\right \rfloor$ for both $i=1,2$, where $k\ge 2$
• Figure 4: A subtree of $T$
• Figure 5: Trees $T$ with $\iota({Mid}(T))=\frac{|V(T)|-2}{2}$

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Claim 1
• Claim 2
• Claim 3
• Claim 4
• Lemma 2.2