Rectilinear drawings of graphs

We consider graphs drawn in the plane such that every edge crosses at most one other edge. We characterize, in terms of two forbidden sub-configurations, which of these graphs are equivalent to drawings such that all edges are straight line segments. As a consequence we obtain a complete characterization of the pairs of dual graphs that can be represented as geometric dual graphs such that all edges except one are straight line segments.     

Colorings and orientations of graphs

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).Research supported in part by a United States-Israel BSF Grant and by a Bergmann Memorial Grant.     

Linear layouts measuring neighbourhoods in graphs

In this paper we introduce the graph layout parameter neighbourhood-width as a variation of the well-known cut-width. The cut-width of a graph G=(V,E) is the smallest integer k, such that there is a linear layout ?:V→{1,…,|V|}, such that for every 1?i<|V| there are at most k edges {u,v} with ?(u)?i and ?(v)>i. The neighbourhood-width of a graph is the smallest integer k, such that there is a linear layout ?, such that for every 1?i<|V| the vertices u with ?(u)?i can be divided into at most k subsets each members having the same neighbourhood with respect to the vertices v with ?(v)>i.We show that the neighbourhood-width of a graph differs from its linear clique-width or linear NLC-width at most by one. This relation is used to show that the minimization problem for neighbourhood-width is NP-complete.Furthermore, we prove that simple modifications of neighbourhood-width imply equivalent layout characterizations for linear clique-width and linear NLC-width.We also show that every graph of path-width k or cut-width k has neighbourhood-width at most k+2 and we give several conditions such that graphs of bounded neighbourhood-width have bounded path-width or bounded cut-width.     

Convex drawings of hierarchical planar graphs and clustered planar graphs

In this paper, we present results on convex drawings of hierarchical graphs and clustered graphs. A convex drawing is a planar straight-line drawing of a plane graph, where every facial cycle is drawn as a convex polygon. Hierarchical graphs and clustered graphs are useful graph models with structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures.We first present the necessary and sufficient conditions for a hierarchical plane graph to admit a convex drawing. More specifically, we show that the necessary and sufficient conditions for a biconnected plane graph due to Thomassen [C. Thomassen, Plane representations of graphs, in: J.A. Bondy, U.S.R. Murty (Eds.), Progress in Graph Theory, Academic Press, 1984, pp. 43–69] remains valid for the case of a hierarchical plane graph. We then prove that every internally triconnected clustered plane graph with a completely connected clustering structure admits a “fully convex drawing,” a planar straight-line drawing such that both clusters and facial cycles are drawn as convex polygons. We also present algorithms to construct such convex drawings of hierarchical graphs and clustered graphs.     

Queue layouts of iterated line directed graphs

In this paper, we study queue layouts of iterated line directed graphs. A k-queue layout of a directed graph consists of a linear ordering of the vertices and an assignment of each arc to exactly one of the k queues so that any two arcs assigned to the same queue do not nest. The queuenumber of a directed graph is the minimum number of queues required for a queue layout of the directed graph.We present upper and lower bounds on the queuenumber of an iterated line directed graph L^k(G) of a directed graph G. Our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the results on the queuenumber of L^k(G), it is shown that for any fixed directed graph G, L^k(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L^k(G). These results are also applied to specific families of iterated line directed graphs such as de Bruijn, Kautz, butterfly, and wrapped butterfly directed graphs. In particular, the queuenumber of k-ary butterfly directed graphs is determined if k is odd.     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.     

Baxter permutations and plane bipolar orientations

We present a simple bijection between Baxter permutations of size n and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations into natural parameters of orientations, and has remarkable symmetry properties. By specialising it to Baxter permutations avoiding the pattern 3142, we obtain a bijection with non-separable planar maps, which had been described only in an implicit recursive manner so far (up to simple symmetries). A further specialization yields a bijection between permutations avoiding 3142 and 2413 and series-parallel maps.     

Homomorphisms of graphs and of their orientations

Consider a set of graphs and all the homomorphisms among them. Change each graph into a digraph by assigning directions to its edges. Some of the homomorphisms preserve the directions and so remain as homomorphisms of the set of digraphs; others do not. We study the relationship between the original set of graph-homomorphisms and the resulting set of digraph-homomorphisms and prove that they are in a certain sense independent. This independence result no longer holds if we start with a proper class of graphs, or if we require that only one direction be given to each edge (unless each homomorphism is invertible, in which case we again prove independence). We also specialize the results to the set consisting of one graph and prove the independence of monoids (groups) of a graph and the corresponding digraph.With 1 Firgure     

Posets and planar graphs

A k-decomposition (G₁,…,G_k) of a graph G is a partition of its edge set to form k spanning subgraphs G₁,…,G_k. The classical theorem of Nordhaus and Gaddum bounds χ(G₁) + χ(G₂) and χ(G₁)χ(G₂) over all 2-decompositions of K_n. For a graph parameter p, let p(k;G) denote the maximum of over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list chromatic number χℓ, and Szekeres–Wilf number σ satisfy ω(2;K_n) = χ(2;K_n) = χℓ(2;K_n) = σ(2;K_n) = n + 1. We obtain lower and upper bounds for ω(k;K_n), χ(k;K_n), χℓ(k;K_n), and σ(k;K_n). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of χ(k;G) over all graphs embedded on a given surface. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Balanced Cayley graphs and balanced planar graphs

A balanced graph is a bipartite graph with no induced circuit of length . These graphs arise in integer linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented. The graphs in this paper are simple.     

-Linked planar graphs

A graph G is (m,n)-linked if for any two disjoint subsets R,BV(G) with |R|m and |B|n, G has two disjoint connected subgraphs containing R and B, respectively. We shall prove that a planar graph with at least six vertices is (3,3)-linked if and only if G is 4-connected and maximal.     

Orienting planar graphs

It is shown that every maximal planar graph (triangulation) can be contracted at an arbitrary point (by identifying it with an adjacent point) so that triangularity is preserved. This is used as a lemma to prove that every triangulation can be (a) oriented so that with three exceptions every point has indegree three, the exceptions having indegrees 0, 1 and 2, and (b) decomposed into three edge-disjoint trees of equal order. The lemma also provides an elementary proof of a theorem of Wagner that every triangulation can be represented by a straight-line drawing.     

Completing orientations of partially oriented graphs

We initiate a general study of what we call orientation completion problems. For a fixed class of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in by orienting the (nonoriented) edges in P. Orientation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically representable graphs. We study orientation completion problems for various classes of oriented graphs, including k‐arc‐strong oriented graphs, k‐strong oriented graphs, quasi‐transitive‐oriented graphs, local tournaments, acyclic local tournaments, locally transitive tournaments, locally transitive local tournaments, in‐tournaments, and oriented graphs that have directed cycle factors. We show that the orientation completion problem for each of these classes is either polynomial time solvable or NP‐complete. We also show that some of the NP‐complete problems become polynomial time solvable when the input‐oriented graphs satisfy certain extra conditions. Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable. The latter generalizes a previous result.