Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S1633 - S1646|
|Published online||02 March 2021|
Almost well-dominated bipartite graphs with minimum degree at least two
Department of Computer Engineering, Gebze Technical University, Kocaeli, Turkey
* Corresponding author: firstname.lastname@example.org
Accepted: 23 April 2020
A dominating set in a graph G = (V, E) is a set S such that every vertex of G is either in S or adjacent to a vertex in S. While the minimum cardinality of a dominating set in G is called the domination number of G denoted by Γ(G), the maximum cardinality of a minimal dominating set in G is called the upper domination number of G denoted by Γ(G). We call the difference between these two parameters the domination gap of G and denote it by µd(G) = Γ(G) − Γ(G). While a graph G with µd(G) = 0 is said to be a well-dominated graph, we call a graph G with µd(G) = 1 an almost well-dominated graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with µd(G) = k, where k ≥ 1, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al. [Ars Comb. 25A (1988) 5–10] imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.
Mathematics Subject Classification: 05C69 / 05C75 / 68R10
Key words: Bipartite graphs / domination gap / well-dominated graphs / almost well-dominated graphs
© EDP Sciences, ROADEF, SMAI 2021
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.