HIMPUNAN DOMINASI TERKENDALI GRAF HASIL OPERASI AMALGAMASI TITIK DAN SISI PADA GRAF SIKLUS BERORDE SAMA

Teori graf merupakan salah satu ilmu terapan matematika yang hingga kini terus dikembangkan. Salah satu konsep yang dipelajari dalam teori graf adalah himpunan dominasi terkendali yang bertujuan untuk menentukan bilangan dominasi terkendali graf hasil operasi amalgamasi titik dan sisi pada graf siklus berorde sama, atau (a(,,)) dan (a(,,,)). Penelitian ini menggunakan metode pembuktian dengan memanfaatkan teorema yang telah ada, yaitu teorema bilangan dominasi terkendali pada graf siklus dan teorema batas atas dan batas bawah bilangan dominasi terkendali. Adapun hasil yang diperoleh, yaitu:  untuk, dan untuk dimana , dan untukdimana

Stratification and domination in graphs.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2006.In a recent manuscript (Stratification and domination in graphs. Discrete Math. 272 (2003), 171-185) a new mathematical framework for studying domination is presented. It is shown that the domination number and many domination related parameters can be interpreted as restricted 2-stratifications or 2-colorings. This framework places the domination number in a new perspective and suggests many other parameters of a graph which are related in some way to the domination number. In this thesis, we continue this study of domination and stratification in graphs. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at the blue vertex v. An F-coloring of a graph G is a red-blue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. The F-domination number yF(GQ of G is the minimum number of red vertices of G in an F-coloring of G. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, we investigate the X-domination number of prisms when X is a 2-stratified 4-cycle rooted at a blue vertex where a prism is the cartesian product Cn x K2, n > 3, of a cycle Cn and a K2. In Chapter 3 we investigate the F-domination number when (i) F is a 2-stratified path P3 on three vertices rooted at a blue vertex which is an end-vertex of the F3 and is adjacent to a blue vertex and with the remaining vertex colored red. In particular, we show that for a tree of diameter at least three this parameter is at most two-thirds its order and we characterize the trees attaining this bound. (ii) We also investigate the F-domination number when F is a 2-stratified K3 rooted at a blue vertex and with exactly one red vertex. We show that if G is a connected graph of order n in which every edge is in a triangle, then for n sufficiently large this parameter is at most (n — /n)/2 and this bound is sharp. In Chapter 4, we further investigate the F-domination number when F is a 2- stratified path P3 on three vertices rooted at a blue vertex which is an end-vertex of the P3 and is adjacent to a blue vertex with the remaining vertex colored red. We show that for a connected graph of order n with minimum degree at least two this parameter is bounded above by (n —1)/2 with the exception of five graphs (one each of orders four, five and six and two of order eight). For n > 9, we characterize those graphs that achieve the upper bound of (n — l)/2. In Chapter 5, we define an f-coloring of a graph to be a red-blue coloring of the vertices such that every blue vertex is adjacent to a blue vertex and to a red vertex, with the red vertex itself adjacent to some other red vertex. The f-domination number yz{G) of a graph G is the minimum number of red vertices of G in an f-coloring of G. Let G be a connected graph of order n > 4 with minimum degree at least 2. We prove that (i) if G has maximum degree A where A 4 with maximum degree A where A 5 with maximum degree A where