ONE OF THE MOST IMPORTANT PROBLEMS IN GRAPH THEORY IS TO FIND THE dominating PARAMETERS WITH MINIMUM CARDINALITY.ALTHOUGH IT IS WELL KNOWN THAT, THESE PARAMETERS ARE NP-HARD. IN THIS NOTE WE WILL STUDY THE CONCEPTS OF TOTAL dominating setS AND TOTAL dominating POLYNOMIALS FROM VIEWPOINT LINEAR PROGRAMMING.

LET G= (V, E) BE A SIMPLE AND UNDIRECTED GRAPH. FOR SOME REAL NUMBER WITH 0<A 1, A set D ÍV IS CALLED AN-dominating set IN G IF EVERY VERTEX U OUTSIDE D HAS AT LEAST A .DU NEIGHBOR (S) IN S WHERE DV IS THE DEGREE OF U.

IN THIS PAPER, WE PROVIDE A SERVICE BY MOVABLE SENSORS IN AD-HOC SENSOR NETWORKS WITHOUT ANY DISCONNECTING IN SERVICE.WE USE dominating set FOR THIS. A dominating set FOR GRAPH G IS A SUBset OF VERTICES SUCH THAT FOR ANY U Î V, EITHER U OR AT LEAST ONE NEIGHBOR OF U IS CONTAINED IN THE SUBset. DUE TO THE LIFETIME IN NODES, AND MOBILITY, THE NETWORKS TOPOLOGY CAN CHANGE OVER THE TIME, SO WE ARE LOOKING FOR SOME SITUATIONS THAT KEEP SERVICE IN THIS UNSAFE SITUATION. WE SHOW THAT IF THE NUMBER OF SERVERS BE AT LEAST [(N+1)/ 3], THEN BY REMOVING NODES FROM NETWORK WE CAN KEEP SERVICE IN WHOLE THE NETWORK WITHOUT ANY PROBLEM.

A set S ÍV IS A CO-SECURE dominating set (CSDS) OF A GRAPH G= (V, E) IF S IS A dominating set, AND FOR EACH U 2 S THERE EXISTS A VERTEX N Î V \ S SUCH THAT UNÎE AND (S\{U}) È {N} A dominating set. THE MINIMUM CARDINALITY OF A CO-SECURE dominating set IN G IS THE CO-SECURE DOMINATION NUMBER ¡CS (G) OF G: IN THIS PAPER WE INITIATE A STUDY OF THIS PARAMETER. WE PROVE THAT THE DECISION PROBLEM FOR THIS PARAMETER IS NP-COMPLETE EVEN WHEN RESTRICTED TO BIPARTITE, CHORDAL OR PLANAR GRAPHS.

THE CONCEPT OF MAJORITY DOMINATION IN GRAPHS HAS BEEN DEFINED IN AT LEAST TWO DIFFERENT WAYS: AS A FUNCTION AND AS A set. IN THIS WORK WE EXTEND THE LATTER CONCEPT TO DIGRAPHS, WHILE WE EXTENDED THE FORMER IN ANOTHER PAPER. GIVEN A DIGRAPH D= (V, A), A set SÍ V IS A MAJORITY OUT-dominating set (MODS) OF D IF (FORMULA). THE MINIMUM CARDINALITY OF A MAJORITY OUT-dominating set IN D IS THE set MAJORITY OUT-DOMINATION NUMBER ¡M+ (D) OF D. IN THIS WORK WE INTRODUCE THESE CONCEPTS AND PROVE SOME RESULTS ABOUT THEM, AMONG WHICH THE CHARACTERIZATION OF MINIMAL MODSS.

A ROMAN ENTIRE dominating FUNCTION ON A GRAPH G= (V, E) IS A FUNCTION H: Z= VÈE → {0, 1, 2} SATISFYING THE CONDITION THAT EACH ELEMENT X ∈ Z FOR WHICH H (X) =0 IS EITHER ADJACENT TO OR INCIDENT WITH AT LEAST ONE ELEMENT Y ∈ Z WITH H (Y) =2. THE WEIGHT OF A ROMAN ENTIRE dominating FUNCTION IS THE VALUE (FORMULA). THE ROMAN ENTIRE DOMINATION NUMBER OF A GRAPH G, DENOTED BY GREN (G), IS THE MINIMUM WEIGHT OF A ROMAN ENTIRE dominating FUNCTION ON G. IN THIS PAPER, WE OBTAIN SEVERAL BOUNDS FOR GREN (G). WE ALSO INVESTIGATE THE BEHAVIOR OF GREN (G) WHEN A VERTEX OR AN EDGE IS DELETED.

LET G= (V, E) BE A SIMPLE GRAPH. A set S &NBSP;Í V IS A dominating set IF EVERY VERTEX IN V \SIS ADJACENT TO AT LEAST ONE VERTEX IN S. THE DOMINATION NUMBER G IS THE MINIMUM CARDINALITY OF A dominating set IN G. A dominating set OF G WITH CARDINALITY G IS CALLED G-set OF G. IN THIS PAPER, WE COUNT THE NUMBER OF G-setS OF CARTESIAN PRODUCT OF G WITH COMPLETE GRAPHS.