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Graphs and Combinatorics

, Volume 27, Issue 5, pp 685–701 | Cite as

Perfect Matchings in Total Domination Critical Graphs

  • Michael A. Henning
  • Anders Yeo
Original Paper

Abstract

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 Gv 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.

Keywords

Total domination Edge-critical Vertex-critical Perfect matching Factor-critical 

Mathematics Subject Classification (2000)

05C69 05C70 

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Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  2. 2.Department of Computer Science, Royal HollowayUniversity of LondonEgham, SurreyUK

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