the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Müller [5]. It also strengthens Schwärzler’s recent proof of one of the open problems of Schrijver’s book [9], abo...

A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for “tripartite” pairings of the nodes, the probability can be computed as a Pfaffian in the en...

An outerplanar graph is a planar graph which can be embedded in the plane such that all nodes lie on the outer face. A series-parallel graph is a graph which is the result of series (subdivision) or parallel (doubling) extensions of the edges of a forest. (Both graph families can be described via families of forbidden minors {K4,K23} and {K4}, respectively, where K4 denotes the complete graph o...

It is well known that planar embeddings of 3-connected graphs are uniquely determined up to isomorphy of the induced complex of nodes, edges and faces of the plane or the 2-sphere [1]. Moreover, each of the isomorphy classes of these embeddings contains a representative that has a convex polygon as outer border and has all edges embedded as straight lines. We fixate the outer polygon of such em...

‎let $g$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎we define an undirected simple graph $delta(g)$ whose‎ ‎vertices are the proper subgroups of $g$, which are not contained in the‎ ‎frattini subgroup of $g$ and two vertices $h$ and $k$ are joined by an edge‎ ‎if and only if $g=langle h‎ , ‎krangle$‎. ‎in this paper we classify finite groups with planar graph‎. ‎...

An outerplanar graph is a graph that admits a planar embedding such that all vertices appear on the boundary of its outer face. Given a positive integer n and a color set C with K > 0 colors, we consider the problem of enumerating all colored and rooted outerplanar graphs with at most n vertices without repetition. We design an efficient algorithm that can generate all required graphs in consta...

Abstract A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V of n vertices in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawing. We give a necessary condition for when pairs of graphs can have a simultaneous embedding with fixed edge...

We give a new proof of the fact that every planar graph is 5choosable, and use it to show that every graph drawn in the plane so that the distance between every pair of crossings is at least 15 is 5-choosable. At the same time we may allow some vertices to have lists of size four only, as long as they are far apart and far from the crossings. Thomassen [5] gave a strikingly beautiful proof that...

‎Let $G$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎We define an undirected simple graph $Delta(G)$ whose‎ ‎vertices are the proper subgroups of $G$, which are not contained in the‎ ‎Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge‎ ‎if and only if $G=langle H‎ , ‎Krangle$‎. ‎In this paper we classify finite groups with planar graph‎. ‎...