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On upper domatic number of graphs

Subhabrata Paul, Kamal Santra

Abstract

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called an \emph{upper domatic partition} of size $k$ if either $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for all $i, j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called the \emph{upper domatic number} of $G$, and it is denoted by $D(G)$. The \textsc{Maximum upper domatic number Problem} involves finding an upper domatic partition of a given graph with the maximum number of parts. It was known that the maximum upper domatic problem can be solved in linear time for trees. In this paper, we prove that this problem can be solved in linear time for \emph{split graphs} and for the \emph{complement of bipartite chain graphs}, two subclasses of chordal graphs. Moreover, we show that this problem can be solved in polynomial time for unicyclic graphs. Finally, we partially solve a conjecture regarding the sink set posed by Haynes et al. [The upper domatic number of a graph, \emph{AKCE Int. J. Graphs Comb.}, 17, 2020].

On upper domatic number of graphs

Abstract

Let be a graph where and are the vertex and edge sets, respectively. For two disjoint subsets and of , we say \textit{dominates} if every vertex of is adjacent to at least one vertex of in . A vertex partition of is called an \emph{upper domatic partition} of size if either dominates or dominates or both for all , where . The maximum integer for which the above partition exists is called the \emph{upper domatic number} of , and it is denoted by . The \textsc{Maximum upper domatic number Problem} involves finding an upper domatic partition of a given graph with the maximum number of parts. It was known that the maximum upper domatic problem can be solved in linear time for trees. In this paper, we prove that this problem can be solved in linear time for \emph{split graphs} and for the \emph{complement of bipartite chain graphs}, two subclasses of chordal graphs. Moreover, we show that this problem can be solved in polynomial time for unicyclic graphs. Finally, we partially solve a conjecture regarding the sink set posed by Haynes et al. [The upper domatic number of a graph, \emph{AKCE Int. J. Graphs Comb.}, 17, 2020].

Paper Structure

This paper contains 12 sections, 16 theorems, 2 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and definition
  3. Upper domatic number in unicyclic graphs
  4. Description of the algorithm
  5. Finding all the maximal $L$-sequences
  6. Running time of the algorithm
  7. Upper domatic number in split graphs
  8. Upper domatic number in the complement of bipartite graphs
  9. Existence of sink set
  10. Graphs with $D(G)=5$
  11. Graphs with $D(G)=6$
  12. Conclusion

Key Result

Lemma 1

Let $G=(V, E)$ be a unicyclic graph, and $\sigma_{u}$ be the ordering described earlier. Also, assume that $c_i$ is any vertex in $\sigma_{u}$ and $N_{G_{[c_i,\sigma_{u}]}}(c_i)=\{v_1^i, v_2^i, \ldots, v_{k_i}^i\}$. Let $L_{\pi}[c_i, G_{[c_i,\sigma_{u}]}] = (l_1, l_2, \ldots, l_{k_i})$ be a maximal

Figures (10)

  • Figure 1: Unicyclic Graph $G$ and the ordering $\sigma_u=\{v_{10}, \ldots, v_{18}, v_1, \ldots, v_4, v_5, \ldots, v_9, u \}$
  • Figure 2: Calculation of X and Y in the tree $\mathcal{T}$
  • Figure 3: Domination among the sets in $\pi$
  • Figure 4: Configuration of $\pi$ in Case \ref{['conj_subcase_1']}
  • Figure 5: Configuration of $\pi$ in Case \ref{['conj_subcase_2']}
  • ...and 5 more figures

Theorems & Definitions (28)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 18 more