Characterization of graphs with orientable total domination number equal to $|V|-1$

Abstract

In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the orientation of such directed graphs valid. The total domination number of $D$, denoted by $γ_t(D)$, is the size of the smallest total dominating set of $D$. For an undirected graph $G$, we investigate the upper (or lower) orientable total domination number of $G$, denoted by $\mathrm{DOM}_t(G)$ (or $\mathrm{dom}_t(G)$), that is the maximum (or minimum) of the total domination numbers over all valid orientations of $G$. We characterize those graphs for which $\mathrm{DOM}_t(G)=|V(G)|-1$, and consequently we show that there exists a family of graphs for which $\mathrm{DOM}_t(G)$ and $\mathrm{dom}_t(G)$ can be as far as possible, namely $\mathrm{DOM}_t(G)=|V(G)|-1$ and $\mathrm{dom}_t(G)=3$.

Figures (9)

• Figure 1: An extremal orientation of a graph with $d^o_+(s)=0$ and $d^o_-(s)\ge 2$
• Figure 2: An extremal orientation of a graph with $d^o_+(s)=0$ and $d^o_-(s)=1$, if there is no vertex that has at least 2 in-neighbors
• Figure 3: An extremal orientation of a graph with $d^o_+(s)=0$ and $d^o_-(s)=1$, if there is a vertex $w_k$ that has at least 2 in-neighbors but all of these in-neighbors belong to $W$
• Figure 4: An extremal orientation of a graph with $d^o_+(s)=0$ and $d^o_-(s)=1$, if there is a vertex $w_k$ that has at least 2 in-neighbors but $N^o_-(w_k)\cap W\ne\emptyset\ne N^o_-(w_k)\cap(V\setminus(W\cup \{s\}))$.
• Figure 5: An extremal orientation of a graph with $d^o_+(s)=0$ and $d^o_-(s)=1$, if there is a vertex $w_k$ that has at least 2 in-neighbors but $N^o_-(w_k)\cap W=\emptyset$.

Theorems & Definitions (7)

• Theorem 1.3
• Proposition 2.2
• proof
• Corollary 2.5
• Corollary 2.6
• proof
• proof : Proof of Theorem \ref{['t:main']} (case analysis)