Geodetic dominating sets and geodetic dominating polynomials of graphs a study

Abstract

Geodetic domination polynomial of a graph is the family of Geodetic domination newlinesets of a graph with cardinality i and dg(G , i) = |Dg(G, i)| .Then the Geodetic newlinedomination polynomial of a graph Dg( G , x) of G is defined as Dg(G, x)= newlineg newline|V(G)| newlinei newlineg newlinei= (G) newlined (G, i)x å , where gg(G) is the geodetic domination number of G. newlineIn chapter 1, we provide the basic definitions and examples which are used in the newlinesubsequent chapters. newlineIn chapter 2 , we introduced geodetic domination polynomials of Complete graph newline(Kn), complete bipartite graph(Km,n), bi-star graph (Bm,n), Star graph ( K1,n), Barbell newlinegraph Bn, Lolipop Graph ( L1,n), Chain triangular Graph (Tn),. newlineIn chapter 3, we studied theGeodetic dominating sets and geodetic dominating newlinepolynomials of path Pn, n³ 2 with n vertices. Let Dg(Pn,i) be the family of geodetic newlinedominating sets of Pn with cardinality i. Then the geodetic domination polynomial Dg( newlinePn, x) of Pn is defined as Dg(Pn, x) = newlineg n newlinen newlinei newlineg n newlinei= (P ) newlineå d (P , i)x where gg(Pn) = newlinen + 2 newline3 newlineé ù newlineêê úú newlineIn chapter 4, we studied theGeodetic dominating sets and geodetic dominating newlinepolynomials of path Cn, n³ 2 with n vertices. Let Dg(Cn,i) be the family of geodetic newlinedominating sets of Cn with cardinality i. Then the geodetic domination polynomial Dg( newlineCn, x) of Cn is defined as Dg(Cn, x) = newlineg n newlinen newlinei newlineg n newlinei= (C ) newlineå d (C , i)x where gg(Cn) = newlinen newline3 newlineé ù newlineêê úú newlineIn chapter 5, we studied the second power of a graph G is a graph with same set newlineof vertices of G and an edge between two vertices if and only if there is a path of length newlineatmost two between them. It is denoted by G2 and is also called square of G. Let newline2 newlineg n D (P , i) be the family of geodetic dominating set of 2 newlinenP with cardinality i. Then the newlinegeodetic domination number of 2 newlinen P is defined as the minimum cardinality taken over all newlinegeodetic dominating sets of vertices in 2 newlinenP and is denoted by gg( 2 newlinenP ),where gg( 2 newlinenP ) = newlinen + 2 newline5 newlineé ù newlineêê úú newline. newlineIn chapter 6, we studied the second power of a graph G is a graph with same set newlineof vertices of G and an edge between two vertices if and only