Let $G$ be a graph in which each vertex initially has weight 1. In step, the unit from $u$ to neighbouring $v$ can moved, provided that on is at least as large $u$. The acquisition number of $G$, denoted by $a_u(G)$, minimum cardinality set vertices with positive end process (over all protocols). this paper, we investigate Erdős-Rényi random $(\mathcal{G}(n,m))_{m =0}^{N}$, where $N = {n \choos...

In this note, we introduce a family of bipartite graphs called path restricted ordered bipartite graphs and present it as an abstract generalization of some well known geometric graphs like unit distance graphs on convex point sets. In the framework of convex point sets, we also focus on a generalized version of Gabriel graphs known as locally Gabriel graphs or LGGs. LGGs can also be seen as th...

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we prov...

It was a long-standing open problem whether the minimum weight dominating set in unit disk graphs has a polynomial-time constant-approximation. In 2006, Ambühl et al solved this problem by presenting a 72-approximation for the minimum weight dominating set and also a 89-approximation for the minimum weight connected dominating set in unit disk graphs. In this paper, we improve their results by ...

Give a graph class G and a nonnegative integer k, we use G+kv, G+ke, and G−ke to denote the classes of graphs that can be obtained from some graph in G by adding k vertices, adding k edges, and deleting k edges, respectively. They are called almost (unit) interval graphs if G is the class of (unit) interval graphs. Almost (unit) interval graphs are well motivated from computational biology, whe...

The intersection graph of a family of arcs on a circle is called a circular-arc graph. This class of graphs admits some interesting subclasses: proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and clique-Helly circular-arc graphs. The intersection graph of a family of chords in a circle is called a circle graph. Analogously, this class of graphs admits some subcla...

The scheduling of task graphs on two identical processors is considered. It is assumed that tasks have unit-execution-time, and arcs are associated with unit-communication-time delays. The problem is to assign the tasks to the two processors and schedule their execution in order to minimize the makespan. A quadratic algorithm is proposed to compute an optimal schedule for a class of series-para...