Liying Kang

Total Restrained Domination in Cubic Graphs

A set S of vertices in a graph Gxa0=xa0(V, E) is a total restrained dominating set (TRDS) of G if every vertex of G is adjacent to a vertex in S and every vertex of V −xa0S is adjacent to a vertex in Vxa0−xa0S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a TRDS of G. Let G be a cubic graph of order n. In this paper we establish an upper bound on γtr(G). If adding the restriction that G is claw-free, then we show that γtr(G)xa0=xa0γt(G) where γt(G) is the total domination number of G, and thus some results on total domination in claw-free cubic graphs are valid for total restrained domination.

Total restrained domination in claw-free graphs

A set S of vertices in a graph G=(V,E) is a total restrained dominating set (TRDS) of G if every vertex of G is adjacent to a vertex in S and every vertex of V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a TRDS of G. In this paper we characterize the claw-free graphs G of order n with γtr(G)=n. Also, we show that γtr(G)≤n−Δ+1 if G is a connected claw-free graph of order n≥4 with maximum degree Δ≤n−2 and minimum degree at least 2 and characterize those graphs which achieve this bound.

Online scheduling on uniform machines with two hierarchies

In this paper we study online scheduling problem on m parallel uniform machines with two hierarchies. The objective is to minimize the maximum completion time (makespan). Machines are provided with different capability. The machines with speed s can schedule all jobs, while the other machines with speed 1 can only process partial jobs. Online algorithms for any 0<s<∞ are provided in the paper. For the case of k=1 and m=2, and the case of some values of s, k=1 and m=3, the algorithms are the best possible, where k is the number of machines with hierarchy 1, and m is the number of machines. Lower bounds for some special cases are also presented.

The p-maxian problem on block graphs

This paper deals with the p-maxian problem on block graphs with unit edge length. It is shown that the two points with maximum distance provide an optimal solution for the 2-maxian problem of block graphs except for K3. It can easily be extended to the p-maxian problem of block graphs. So we solve the p-maxian problem on block graphs in linear time.

Matching Properties in Total Domination Vertex Critical Graphs

A vertex subset S of a graph Gxa0=xa0(V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γt(G−v)xa0<xa0γt(G). A total domination vertex critical graph G is called k-γt-critical if γt(G)xa0=xa0k. In this paper we first show that every 3-γt-critical graph G of even order has a perfect matching if it is K1,5-free. Secondly, we show that every 3-γt-critical graph G of odd order is factor-critical if it is K1,5-free. Finally, we show that G has a perfect matching if G is a K1,4-free 4-γt(G)-critical graph of even order and G is factor-critical if G is a K1,4-free 4-γt(G)-critical graph of odd order.

Matching and domination numbers in r-uniform hypergraphs

A matching is a set of pairwise disjoint hyperedges of a hypergraph H. The matching number

Further properties on the degree distance of graphs

nu (H)

Clique-Coloring Claw-Free Graphs

ν(H) of H is the maximum cardinality of a matching. A subset D of vertices of H is called a dominating set of H if for every vertex v not in D there exists