Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl [1], in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction...

We define the intersection complex for universal cover of a compact weakly special square and show that it is quasi-isometry invariant. By using this invariant, we study quasi-isometric classification 2-dimensional right-angled Artin groups planar graph 2-braid groups. Our results two well-known cases groups: (1) those whose defining graphs are trees (2) outer automorphism finite. Finally, ther...

This paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an eff...

We give an O(nd + n log n) algorithm computing the number of minimum (s, t)-cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest s-t path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomial-time algorithm for unweighted planar graphs with both s and t lying on the outer face. Our results hold for all locations of ...

Given an edge-weighted undirected graph and a list of k source-sink pairs of vertices, the well-known minimum multicut problem consists in selecting a minimum-weight set of edges whose removal leaves no path between every source and its corresponding sink. We give the first polynomial-time algorithm to solve this problem in planar graphs, when k is fixed. Previously, this problem was known to r...

An outer-1-planar graph is a admitting drawing in the plane so that all vertices appear outer region of and every edge crosses at most one other edge. This paper establishes local structure graphs by proving each contains seventeen fixed configurations, list those configurations minimal sense for configuration there exist containing this do not contain any another sixteen configurations. There ...

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...

Monotone Drawings of Graphs p. 13 Upward Geometric Graph Embeddings into Point Sets p. 25 On a Tree and a Path with No Geometric Simultaneous Embedding p. 38 Difference Map Readability for Dynamic Graphs p. 50 Maximizing the Total Resolution of Graphs p. 62 Plane Drawings of Queue and Deque Graphs p. 68 An Experimental Evaluation of Multilevel Layout Methods p. 80 Orthogonal Graph Drawing with ...

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

We consider the problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable—indeed, solvable in linea...