Denote by σ̄k = min {d(x1)+d(x2)+ · · ·+d(xk)−|N(x1)∩N(x2)∩· · ·∩N(xk)| | x1, x2, · · · , xk are k independent vertices in G}. Let n and m denote the number of vertices and edges of G. For any connected graph G, we give a polynomial algorithm in O(nm) time to either find two disjoint paths P1 and P2 such that |P1|+ |P2| ≥ min{σ̄4, n} or output G = ∪i=1Gi such that for any i, j ∈ {1, 2, ..., k} (k...

Let IL be the set of all hereditary and additive properties of graphs. For P1,P2 ∈ IL, the reducible property R = P1 ◦ P2 is defined as follows: G ∈ R if and only if there is a partition V (G) = V1 ∪ V2 of the vertex set of G such that 〈V1〉G ∈ P1 and 〈V2〉G ∈ P2. The aim of this paper is to investigate the structure of the reducible properties of graphs with emphasis on the uniqueness of the dec...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

ABSTRACT Let ‎G=(V,E) ‎be a‎ ‎simple ‎connected ‎graph ‎with ‎vertex ‎set ‎V‎‎‎ ‎and ‎edge ‎set ‎‎‎E. ‎The Szeged index ‎of ‎‎G is defined by ‎ where ‎ respectively ‎ ‎ is the number of vertices of ‎G ‎closer to ‎u‎ (‎‎respectively v)‎ ‎‎than ‎‎‎v (‎‎respectively u‎).‎ ‎‎If ‎‎‎‎S ‎is a‎ ‎set ‎of ‎size‎ ‎ ‎ ‎let ‎‎V ‎be ‎the ‎set ‎of ‎all ‎subsets ‎of ‎‎S ‎of ‎size ‎3. ‎Then ‎we ‎define ‎t...

The one-round discrete Voronoi game, with respect to a n-point user set U , consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set of facilities F1 following which P2 chooses another set of facilities F2, disjoint from F1. The payoff of P2 is defined as the cardinality of the set of points in U which are closer to a facility in F2 than to every facility in F1, and t...

A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

Let G = (V,E) be a connected undirected graph. For any vertex v ∈ V , the closed neighborhood of v is N [v] = {v} ∪ {u ∈ V | uv ∈ E }. For S ⊆ V , the closed neighborhood of S is N [S] = ⋃ v∈S N [v]. The subgraph weakly induced by S is 〈S〉w = (N [S], E ∩ (S × N [S])). A set S is a weakly-connected dominating set of G if S is dominating and 〈S〉w is connected. The weakly-connected domination numb...

A subset S of V is called a total dominating set if every vertex in V is adjacent to some vertex in S. The total domination number γt (G) of G is the minimum cardinality taken over all total dominating sets of G. A dominating set is called a connected dominating set if the induced subgraph 〈S〉 is connected. The connected domination number γc(G) of G is the minimum cardinality taken over all min...

Constructing a connected dominating set as the virtual backbone plays an important role in wireless networks. In this paper, we propose two novel approximate algorithms for dominating set and connected dominating set in wireless networks, respectively. Both of the algorithms are based on edge dominating capability which is a novel notion proposed in this paper. Simulations show that each of pro...

The one-round discrete Voronoi game, with respect to a n-point user set U , consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set of facilities F1 following which P2 chooses another set of facilities F2, disjoint from F1. The payoff of P2 is defined as the cardinality of the set of points in U which are closer to a facility in F2 than to every facility in F1, and t...