Perfect Matchings in Total Domination Critical Graphs
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A graph is total domination edge-critical if the addition of any edge decreases the total domination number, while a graph with minimum degree at least two is total domination vertex-critical if the removal of any vertex decreases the total domination number. A 3 t EC graph is a total domination edge-critical graph with total domination number 3 and a 3 t VC graph is a total domination vertex-critical graph with total domination number 3. A graph G is factor-critical if G − v has a perfect matching for every vertex v in G. In this paper, we show that every 3 t EC graph of even order has a perfect matching, while every 3 t EC graph of odd order with no cut-vertex is factor-critical. We also show that every 3 t VC graph of even order that is K 1,7-free has a perfect matching, while every 3 t VC graph of odd order that is K 1,6-free is factor-critical. We show that these results are tight in the sense that there exist 3 t VC graphs of even order with no perfect matching that are K 1,8-free and 3 t VC graphs of odd order that are K 1,7-free but not factor-critical.
KeywordsTotal domination Edge-critical Vertex-critical Perfect matching Factor-critical
Mathematics Subject Classification (2000)05C69 05C70
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- 11.Haynes, T.W., Henning, M.A., van der Merwe, L.C., Yeo, A.: On a conjecture of Murty and Simon on diameter two critical graphs. Manuscript, July 2009Google Scholar
- 12.Haynes, T.W, Henning, M.A., van der Merwe, L.C., Yeo, A.: On the existence of k-partite or K p-free total domination edge-critical graphs. Discrete Math. (to appear)Google Scholar
- 19.Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)Google Scholar
- 21.Shan, E.: Personal communicationGoogle Scholar
- 22.Simmons, J.: Closure operations and hamiltonian properties of independent and total domination critical graphs. Ph.D. Thesis. PhD advisor: Gary MacGillvray. University of Victoria (2005)Google Scholar
- 32.van der Merwe, L.C.: Total domination edge critical graphs. Ph.D. Thesis. University of South Africa (1999)Google Scholar