In a graph, vertex dominates itself and its neighbors. A subset S of vertices graph G is double dominating set if every at least twice. The domination number γ ×2 ( ) the minimum cardinality . this paper, we prove that maximal outerplanar order n bounded above by + k /2, where pairs consecutive degree two with distance 3 on outer cycle. We also ×2( ≤ 5 /8 for Hamiltonian planar ≥ 7.

It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. The latter extends a theorem of H. Whitney and gives two-page book embeddings for X-trees and square grids.

We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordere...

Colin de Verdi ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n ? 4, and if it is planar, then (G) n ? 5. We give a full characterization of maximal planar graphs with (G) = n ? 5. In the opposite di...

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth k, then we can triangulate it so that the resulting graph has pathwidth O(k) (w...

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97–108] that there are no 5-connected planar w...