Signed mixed dominating functions in complete bipartite graphs

Abstract

Let G=(V, E) be a simple graph with vertex set V and edge set E. A signed mixed dominating function (SMDF) of G is a function f: V∪E→{−1, 1} such that for every element x∈V∪E, where N_m(x) is the set, called mixed neighbourhood of x, of elements of V∪E adjacent or incident to x. In other words, for every list-assignment of two colours {−1, 1} to every elements of V∪E, there is a list-colouring of vertices and edges of G such that all mixed neighbourhoods contain more 1′s than−1′s. The weight of f is w(f)=∑_x∈V∪Ef(x). The signed mixed domination number γ_s*(G) of G is the minimum weight of all possible SMDF of G. In this paper, we determine the exact value of the signed mixed domination number in a complete bipartite graph.

Keywords:

• signed domination
• signed mixed dominating function
• signed mixed domination number
• complete bipartite graph

2010 AMS Subject Classifications::

• 05C69
• 05A10

Acknowledgement

Research was partially supported by the National Nature Science Foundation of China (No. 11171207) and the Scientific Project for the Training of ‘333’ High-Level Talents in Jiangsu Province.