The Complexity of Computing Signed (Total) Domatic Numbers of Graphs

Rui Li1, Hongyu Liangt2
1Jiangxi College of Applied Technology, Ganzhou 341000, China.
2Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China

Abstract

Let \( G = (V, E) \) be a graph. A function \( f: V \to \{-1, 1\} \) is called a signed dominating function on \( G \) if \( \sum_{u \in N_G[v]} f(u) \geq 1 \) for each \( v \in V \), where \( N_G[v] \) is the closed neighborhood of \( v \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed dominating functions on \( G \) is called a signed dominating family (of functions) on \( G \) if \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V \). The signed domatic number of \( G \) is the maximum number of functions in a signed dominating family on \( G \). The signed total domatic number is defined similarly, by replacing the closed neighborhood \( N_G[v] \) with the open neighborhood \( N_G(v) \) in the definition. In this paper, we prove that the problems of computing the signed domatic number and the signed total domatic number of a given graph are both NP-hard, even if the graph has bounded maximum degree. To the best of our knowledge, these are the first NP-hardness results for these two variants of the domatic number.