Abstract
Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a restrained dominating set if every vertex in \(V-S\) is adjacent to a vertex in \(S\) and to a vertex in \(V-S\). The restrained domination number of \(G\), denoted \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). Consider a bipartite graph \(G\) of order \(n\ge 4,\) and let \(k\in \{2,3,...,n-2\}.\) In this paper we will show that if \(\gamma _{r}(G)=k\), then \(m\le ((n-k)(n-k+6)+4k-8)/4\). We will also show that this bound is best possible.
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References
Arumugam S, Velammal S (1999) Maximum size of a connected graph with given domination parameters. Ars Comb 52:221–227
Dankelmann P, Domke GS, Goddard W, Grobler P, Hattingh JH, Swart HC (2004) Maximum sizes of graphs with given domination parameters. Discret Math 281:137–148
Domke GS, Hattingh JH, Hedetniemi ST, Laskar RC, Markus LR (1999) Restrained domination in graphs. Discret Math 203:61–69
Domke GS, Hattingh JH, Henning MA, Markus LR (2000) Restrained domination in graphs with minimum degree two. J Combin Math Combin Comput 35:239–254
Ferneyhough S, Haas R, Hanson D, MacGillivray G (2002) Star forests, dominating sets and Ramsey-type problems. Discret Math 245:255–262
Hattingh JH, Jonck E, Joubert EJ, Plummer AR (2008) Nordhaus–Gaddum results for restrained domination and total restrained domination in graphs. Discret Math 308:1080–1087
Haynes TW, Hedetniemi ST, Slater PJ (1997a) Fundamentals of domination in graphs. Marcel Dekker, New York
Haynes TW, Hedetniemi ST, Slater PJ (1997b) Domination in graphs: advanced topics. Marcel Dekker, New York
Joubert EJ (2013) Maximum sizes of graphs with given restrained domination numbers. Discret Appl Math 161:829–837
Telle JA, Proskurowski A (1997) Algorithms for vertex partitioning problems on partial k-trees. SIAM J Discret Math 10:529–550
Zelinka B (2005) Remarks on restrained and total restrained domination in graphs. Czechoslovak Math J 55(130):165–173
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Joubert, E.J. An inequality that relates the size of a bipartite graph with its order and restrained domination number. J Comb Optim 31, 44–51 (2016). https://doi.org/10.1007/s10878-014-9709-1
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DOI: https://doi.org/10.1007/s10878-014-9709-1