We give two results on domination in graphs, including a proof of a conjecture of Favaron, Henning, Mynhart and Puech [2]. Corollary 2 was found by four separate subsets of the authors. We decided to give this joint presentation of our results. We first offer a result about bipartite graphs.Lemma 1 Let G be a bipartite graph with partite sets (X, Y ) whose vertices in Y are of minimum degree at least 3. Then there exists a set A ⊂ X of size at most |X ∪ Y |/4 such that every vertex in Y is adjacent to a vertex in A.Proof: The proof is by induction on |V (G)| + |E(G)|. The smallest graph as described in the lemma is K 1,3 , for which the statement holds. This gives the start of our induction. Let x = |X| and y = |Y |. If there exists a vertex v in Y of degree at least 4, then delete any edge e incident to v. The subset A of G − e guaranteed by the inductive hypothesis is adjacent in G to every vertex in Y as desired. So we may assume that the vertices in Y are all of
Applying the recently obtained distributive lattice structure on the set of 1-factors, we show that the resonance graphs of any benzenoid systems G, as well as of general plane (weakly) elementary bipartite graphs, are median graphs and thus extend greatly Klavžar et al.'s result. The n-dimensional vectors of nonnegative integers as a labelling for the 1-factors of G with n inner faces are described. The labelling preserves the partial ordering of the above-mentioned lattice and can be transformed into a binary coding for the 1-factors. A simple criterion for such a labelling being binary is given. In particular, Klavžar et al.'s algorithm is modified to generate this binary coding for the 1-factors of a cata-condensed benzenoid system.
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