Let G=(V,E) be a simple graph with vertex set V and edge set E. A subset W⊆V∪E is a mixed dominating set if every element x∈(V∪E)\W is either adjacent or incident to an element of W. The mixed domination problem is to find a minimum mixed dominating set of G. In this paper we first prove that a connected graph is a tree if and only if its total graph is strongly chordal, and thus we obtain a polynomial-time algorithm for this problem in trees. Further we design another linear-time labeling algorithm for this problem in trees. At the end of the paper, we show that the mixed domination problem is NP-complete even when restricted to split graphs, a subclass of chordal graphs. © 2010 Elsevier B.V. All rights reserved.
Zhao, Y., Kang, L., & Sohn, M. Y. (2011). The algorithmic complexity of mixed domination in graphs. Theoretical Computer Science, 412(22), 2387–2392. https://doi.org/10.1016/j.tcs.2011.01.029