Graphs and Combinatorics

, Volume 30, Issue 3, pp 647–660 | Cite as

Subset-Sum Representations of Domination Polynomials

Original Paper

Abstract

The domination polynomial D(G, x) is the ordinary generating function for the dominating sets of an undirected graph G = (V, E) with respect to their cardinality. We consider in this paper representations of D(G, x) as a sum over subsets of the edge and vertex set of G. One of our main results is a representation of D(G, x) as a sum ranging over spanning bipartite subgraphs of G. Let d(G) be the number of dominating sets of G. We call a graph G conformal if all of its components are of even order. Let Con(G) be the set of all vertex-induced conformal subgraphs of G and let k(G) be the number of components of G. We show that
$$d(G) = \sum \limits_{H\in{\rm Con}(G)}2^{k(H)}$$
.

Keywords

Domination polynomial Dominating set Graph polynomial 

Mathematics Subject Classification (2000)

05C30 05C31 05C69 

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Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsCape Breton UniversitySydneyCanada
  3. 3.Faculty of Mathematics, Sciences, and Computer ScienceHochschule Mittweida - University of Applied SciencesMittweidaGermany

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