The Wiener index is one of the oldest graph parameter which is used to study molecular-graph-based structure. This parameter was first proposed by Harold Wiener in 1947 to determining the boiling point of paraffin. The Wiener index of a molecular graph measures the compactness of the underlying molecule. This parameter is wide studied area for molecular chemistry. It is used to study the physio...

We show that there is an O(n3) algorithm to approximate the bandwidth of an AT-free graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + n logn) algorithm to approximate the bandwidth of an AT-free graph within a factor 4 and anO(n+ e) algorithm with a factor 6. For the special cases of permutation graphs and trapezoid gr...

We show that the minimum fill-in and the minimum interval graph completion of a d-trapezoid graph can be computed in time O(n). We also show that the treewidth and the pathwidth of a d-trapezoid graph can be computed by an O(n tw(G) ) time algorithm. For both algorithms, d is supposed to be a fixed positive integer and it is required that a suitable intersection model of the given d-trapezoid g...

A Hamiltonian path (cycle) of a graph is a simple path (cycle) which visits each vertex of the graph exactly once. The Hamiltonian path (cycle) problem is to determine whether a graph contains a Hamiltonian path (cycle). A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. Supergrid graphs were first introduced by us and include grid grap...

We study on-line colorings of certain graphs given as intersection graphs of objects “between two lines”, i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axisaligned recta...

For a graph G=(V,E), subset D of vertex set V, is dominating G if every not in adjacent to at least one D. A with no isolated vertices called paired (PD-set), the subgraph induced by has perfect matching. The Min-PD problem requires compute PD-set minimum cardinality. decision version NP-complete even when belongs restricted classes such as bipartite graphs, chordal graphs etc. On other side, e...