Yee-Hock Peng

Dominating Sets and Domination Polynomials of Paths

Let be a simple graph. A set is a dominating set of , if every vertex in is adjacent to at least one vertex in . Let be the family of all dominating sets of a path with cardinality , and let . In this paper, we construct , and obtain a recursive formula for . Using this recursive formula, we consider the polynomial , which we call domination polynomial of paths and obtain some properties of this polynomial.

The chromaticity of s -bridge graphs and related graphs

Abstract The graph consisting of s paths joining two vertices is called s -bridge graph. In this paper, we discuss the chromaticity of some families of s -bridge graphs, especially 4-bridge graphs, and some graphs related to s -bridge graphs.

On the chromatic uniqueness of certain bipartite graphs

Let K(p, q), p ⩽ q, denote the complete bipartite graph in which the two partite sets consist of p and q vertices, respectively. We denote by K-r(p, q), the family of all graphs obtained by deleting any r distinct edges from K(p, q). Teo and Koh showed that K-1(p, q) is chromatically unique (in short χ-unique) for all p, q such that 3 ⩽ p ⩽ q. In this paper, we obtain a sufficient condition for a graph in K-2(p, q) to be χ-unique. Using this result, we then prove that each graph in K-2(p, p + d) is χ-unique for p ⩾ 4and 0 ⩽ d ⩽ 3. For d ⩾ 4, the graphs in K-2(p, q + d) are χ-unique if p > (A + √B)/4d2, where A and B are polynomials in d. We also show that each graph (≇K(4, 4) – K(1, 3)) in K-3(p, p + d) is χ-unique, for p ⩾ 4 and d=0, 1; and all graphs in K-3(p, p + 2) are χ-unique if and only if all graphs in K-4(p + 1, p + 1) are χ-unique, where p ⩾ 4. Finally we prove that every graph in K-4(p, p + 1) is χ-unique for p ⩾ 5.

Dominating sets of centipedes

Abstract Let G = (V, E) be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V − S is adjacent to at least one vertex in S. Let be the family of all dominating sets of a graph G with cardinality i, and G* be the graph obtained by appending a single pendant edge to each vertex of graph G. We call a centipede, where Pn is a path with n vertices. In this paper we study the dominating sets of centipedes and obtain the number of dominating sets of . We show that , i), where Cn and Gn are respectively, cycle and arbitrary graph of order n.

Chromatic equivalence classes of certain cycles with edges

Abstract Let P(G) denote the chromatic polynomial of a graph G . Two graphs G and H are chromatically equivalent, written G∼H , if P(G)=P(H) . A graph G is chromatically unique if for any graph H , G∼H implies that G is isomorphic with H . In this paper, we give the necessary and sufficient conditions for a family of generalized polygon trees to be chromatically unique.

Construction of Dominating Sets of Certain Graphs

Let be a simple graph. A set is a dominating set of , if every vertex in is adjacent to at least one vertex in . We denote the family of dominating sets of a graph with cardinality by . In this paper we introduce graphs with specific constructions, which are denoted by . We construct the dominating sets of by dominating sets of graphs , , and . As an example of , we consider . As a consequence, we obtain the recursive formula for the number of dominating sets of .

On the chromaticity of complete multipartite graphs with certain edges added

Let P(G,@l) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,@l)=P(G,@l) implies H is isomorphic to G. For integers k>=0, t>=2, denote by K((t-1)xp,p+k) the complete t-partite graph that has t-1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((t-1)xp,p+k) by adding a set S of s edges to the partite set of size p+k such that is bipartite. If s=1, denote the only graph in K(s,t,p,k) by K^+((t-1)xp,p+k). In this paper, we shall prove that for k=0,1 and p+k>=s+2, each graph G@?K(s,t,p,k) is chromatically unique if and only if is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K^+((t-1)xp,p+k) is chromatically unique for k=0,1 and p+k>=3.