Rectangular drawings of plane graphs without designated corners

This paper addresses the problem of finding rectangular drawings of plane graphs, in which each vertex is drawn as a point, each edge is drawn as a horizontal or a vertical line segment, and the contour of each face is drawn as a rectangle. A graph is a 2–3 plane graph if it is a plane graph and each vertex has degree 3 except the vertices on the outer face which have degree 2 or 3. A necessary and sufficient condition for the existence of a rectangular drawing has been known only for the case where exactly four vertices of degree 2 on the outer face are designated as corners in a 2–3 plane graph G. In this paper we establish a necessary and sufficient condition for the existence of a rectangular drawing of G for the general case in which no vertices are designated as corners. We also give a linear-time algorithm to find a rectangular drawing of G if it exists.     

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.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

Characterizations of bipartite and Eulerian partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.     

Resistance characterizations of equiarboreal graphs

A weighted (unweighted) graph $G$ is called equiarboreal if the sum of weights (the number) of spanning trees containing a given edge in $G$ is independent of the choice of edge. In this paper, we give some resistance characterizations of equiarboreal weighted and unweighted graphs, and obtain the necessary and sufficient conditions for $k$-subdivision graphs, iterated double graphs, line graphs of regular graphs and duals of planar graphs to be equiarboreal. Applying these results, we obtain new infinite families of equiarboreal graphs, including iterated double graphs of 1-walk-regular graphs, line graphs of triangle-free 2-walk-regular graphs, and duals of equiarboreal planar graphs.     

Infinite matroids in graphs

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals.In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.     

Supervisory Control of Extended Timed Event Graphs

This paper describes the dynamic behavior of extended timed event graphs related to place delay in the dioid framework. By Cofer and Garg‘s supervisory control theory, we address control problems of extended timed event graphs. Supervisory control of extended timed event graphs (a class of discrete event dynamic systems) is studied in the dioid framework, a necessary and sufficient condition for the ideals of the set of firing time sequences of transitions to be controllable is presented. We prove all the strongly controllable subsets can form a complete lattice,     

Eulerian partial duals of plane graphs

In the paper Combinatorica 33(2) (2013) 231–252, Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. An open problem posed in their paper is the characterization of Eulerian partial duals of plane graphs. In this article, we solve this problem by considering half‐edge orientations of medial graphs.     

Crossing minimization in extended level drawings of graphs

The most popular method of drawing directed graphs is to place vertices on a set of horizontal or concentric levels, known as level drawings. Level drawings are well studied in Graph Drawing due to their strong application for the visualization of hierarchy in graphs. There are two drawing conventions: Horizontal drawings use a set of parallel lines and radial drawings use a set of concentric circles.In level drawings, edges are only allowed between vertices on different levels. However, many real world graphs exhibit hierarchies with edges between vertices on the same level. In this paper, we initiate the new problem of extended level drawings of graphs, which was addressed as one of the open problems in social network visualization, in particular, displaying centrality values of actors. More specifically, we study minimizing the number of edge crossings in extended level drawings of graphs. The main problem can be formulated as the extended one-sided crossing minimization problem between two adjacent levels, as it is folklore with the one-sided crossing minimization problem in horizontal drawings.We first show that the extended one-sided crossing minimization problem is NP-hard for both horizontal and radial drawings, and then present efficient heuristics for minimizing edge crossings in extended level drawings. Our extensive experimental results show that our new methods reduce up to 30% of edge crossings.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

A full characterization of invariant embeddability of unimodular planar graphs

When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.     

Dominating set games

In this paper, we study cooperative cost games arising from domination problems on graphs. We introduce three games to model the cost allocation problem and we derive a necessary and sufficient condition for the balancedness of all three games.     

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.     

The uniquely embeddable planar graphs

Whitney [7] proved in 1932 that for any two embeddings of a planar 3-connected graph, their combinatorial duals are isomorphic. In this manner, the term “uniquely embeddable planar graph” was introduced. It is a well-known fact that combinatorial and geometrical duals are equivalent concepts. In this paper, the concept of unique embeddability is introduced in terms of special types of isomorphisms between any two embeddings of a planar graph. From this, the class U of all graphs which are uniquely embeddable in the plane according to this definition, is determined, and the planar 3-connected graphs are a proper subset of U. It turns out that the graphs in U have a unique geometrical dual (i.e., for any two embeddings of such a graph, their geometrical duals are isomorphic). Furthermore, the theorems and their proofs do not involve any type of duals.     

Cycles with Prescribed and Forbidden Sets of Elements in Cubic Graphs

 A necessary and sufficient condition for the existence of a cycle containing a set of three vertices and an edge excluding another edge is obtained for 3-connected cubic graphs by means of canonical contractions. A necessary and sufficient condition is also obtained for a cyclically 4-connected cubic graph to have a cycle that contains a set of four vertices and that avoids another vertex. Received: October 4, 1999 Final version received: June 14, 2000     

Drawings of planar graphs with few slopes and segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.     

正规Tiling的邻居系

该文借助Tiling的拓扑图和边邻居图, 给出了正规Tiling 中两个Tiles 拥有相同的邻居系的一个充分必要条件, 并利用此条件证明了正规Tiling 中不可能存在三个不同的Tiles 具有相同的邻居系, 从而在一定程度上回答了Grünbaum 的一个公开问题.     

Convex Quadratic Programming Approach to the Maximum Matching Problem

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.     

The Hamiltonian connectivity of rectangular supergrid graphs

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as subgraphs. The Hamiltonian path problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that the Hamiltonian path problem for supergrid graphs is also NP-complete. The Hamiltonian paths on supergrid graphs can be applied to compute the stitching traces of computer sewing machines. Rectangular supergrid graphs form a popular subclass of supergrid graphs, and they have strong structure. In this paper, we provide a constructive proof to show that rectangular supergrid graphs are Hamiltonian connected except one trivial forbidden condition. Based on the constructive proof, we present a linear-time algorithm to construct a longest path between any two given vertices in a rectangular supergrid graph.     

Convex Quadratic Programming Approach to the Maximum Matching Problem

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.