A rectangular drawing of a planar graph G is in which vertices are mapped to grid points, edges horizontal and vertical straight-line segments, faces drawn as rectangles. Sometimes this latter constraint relaxed for the outer face. In paper, we study drawings have unit length. We show complexity dichotomy problem deciding existence unit-length drawing, depending on whether face must also be rec...

We show that every n-vertex planar graph admits a simultaneous embedding with no mapping and with fixed edges with any (n/2)-vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous...

This paper is dedicated to propose an algorithm in order to generate the certain isomers of some well-known fullerene bases. Furthermore, we enlist the bipartite edge frustration correlated with some of symmetrically distinct innite families of fullerenes generated by the oered process.

We show that every n-vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any (n/2)vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous ...

This paper presented study on convex drawing of planar graph. In graph theory, a planar graph is a graph that can be embedded in the plane. A planar graph is one that can be drawn on a plane in such a way that there are no “edge crossings,” i.e. edges intersect only at their common vertices. Convex polygon has all interior angles less than or equal to 180°. A graph is called a convex drawing if...

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] gave such a graph which is not 3-choosable. We prove that every triangle-free planar graph such that 4-cycles do not share edges with other 4and 5-cycles is 3-choosable. T...

Given a planar graph G, what is the maximum number of collinear vertices in a planar straight-line drawing of G? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known: Every n-vertex planar graph has a planar straight-line drawing with Ω( √ n) collinear vertices; for every n, there...

let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. We show that every 1-planar drawing of any 1-planar graph on n vertices has at most n − 2 crossings; moreover, this bound is tight. By this novel necessary condition for 1-planarity, we characterize the 1-planarity of Cartesian product Km × Pn. Based on this condition, we a...

Given an edge-weighted graph G and a list of source-sink pairs of terminal vertices of G, the minimum multicut problem consists in selecting a minimum weight set of edges of G whose removal leaves no path from the ith source to the ith sink, for each i. Few tractable special cases are known for this problem. In this paper, we give a simple polynomial-time algorithm solving it in undirected plan...