摘要
A set S of vertices in a graph G = （V, E） without isolated vertices is a total outer-connected dominating set （TCDS） of G if S is a total dominating set of G and G[V - S] is connected. The total outer-connected domination number of G, denoted by γtc（G）, is the minimum cardinality of a TCDS of G. For an arbitrary graph without isolated vertices, we obtain the upper and lower bounds on γtc（G） ＋ γytc（G）, and characterize the extremal graphs achieving these bounds.
A set S of vertices in a graph G = （V, E） without isolated vertices is a total outer-connected dominating set （TCDS） of G if S is a total dominating set of G and G[V - S] is connected. The total outer-connected domination number of G, denoted by γtc（G）, is the minimum cardinality of a TCDS of G. For an arbitrary graph without isolated vertices, we obtain the upper and lower bounds on γtc（G） ＋ γytc（G）, and characterize the extremal graphs achieving these bounds.
基金
Supported by National Natural Science Foundation of China （Grant Nos. 60773078, 10971131） and Shanghai Leading Academic Discipline Project （Grant No. S30104） Thank the referees sincerely for all of the helpful suggestions.
