Classifying cubic symmetric graphs of order 10p or 10p~2

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ≥ 1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s-regular cubic graphs of order 10p or 10p2 are also classified for each s ≥ 1 and each prime p, of which the proof depends on the classification of finite simple groups.     

The ratio of the numbers of odd and even cycles in outerplanar graphs

In this paper, we investigate the ratio of the numbers of odd and even cycles in outerplanar graphs. We verify that the ratio generally diverges to infinity as the order of a graph diverges to infinity. We also give sharp estimations of the ratio for several classes of outerplanar graphs, and obtain a constant upper bound of the ratio for some of them. Furthermore, we consider similar problems in graphs with some pairs of forbidden subgraphs/minors, and propose a challenging problem concerning claw-free graphs.     

Tetravalent one-regular graphs of order 2<Emphasis Type="Italic">pq</Emphasis>

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this article a complete classification of tetravalent one-regular graphs of order twice a product of two primes is given. It follows from this classification that with the exception of four graphs of orders 12 and 30, all such graphs are Cayley graphs on Abelian, dihedral, or generalized dihedral groups.     

Partial Characterizations of 1‐Perfectly Orientable Graphs

We study the class of 1‐perfectly orientable graphs, that is, graphs having an orientation in which every out‐neighborhood induces a tournament. 1‐perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this article, we develop several results on 1‐perfectly orientable graphs. In particular, we (i) give a characterization of 1‐perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1‐perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1‐perfectly orientable graphs, and (iv) characterize the class of 1‐perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1‐perfectly orientable cobipartite graphs coincides with the class of cobipartite circular arc graphs.     

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.     

A note on immersion minors and planarity

Graph minors play an important role in graph theory. The focus of this paper is on immersion minors and their relationship to planarity. In general, planar graphs can have non-planar immersion minors. This paper shows that by placing a simple restriction on the immersion-minor operations, all immersion minors of a planar graph are planar. This then allows one to easily obtain a characterization of planar graphs using immersion minors. A dual form of this characterization, as well as an extension to binary matroids, are also considered.     

Hereditary Efficiently Dominatable Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C₄)‐free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw‐free graphs.     

On the existence of special depth first search trees

The Depth First Search (DFS) algorithm is one of the basic techniques that is used in a very large variety of graph algorithms. Most applications of the DFS involve the construction of a depth-first spanning tree (DFS tree). In this paper, we give a complete characterization of all the graphs in which every spanning tree is a DFS tree. These graphs are called Total-DFS-Graphs. We prove that Total-DFS-Graphs are closed under minors. It follows by the work of Robertson and Seymour on graph minors that there is a finite set of forbidden minors of these graphs and that there is a polynomial algorithm for their recognition. We also provide explicit characterizations of these graphs in terms of forbidden minors and forbidden topological minors. The complete characterization implies explicit linear algorithm for their recognition. In some problems the degree of some vertices in the DFS tree obtained in a certain run are crucial and therefore we also consider the following problem: Let G = (V,E) be a connected undirected graph where |V| = n and let d ? Nⁿ be a degree sequence upper bound vector. Is there any DFS tree T with degree sequence d _T that violates d (i.e., d _T ≤ d , which means: E j such that d _T(j) > d (j))? We show that this problem is NP-complete even for the case where we restrict the degree of only on specific vertex to be less than or equal to k for a fixed k ≥ 2 (i.e., d = (n - 1, ?, n - 1, k, n - 1, ?, n - 1)). 0 1995 John Wiley & Sons, Inc.     

Edge-primitive Cayley graphs on abelian groups and dihedral groups

A graph is called edge-primitive if its automorphism group acts primitively on its edge set. In 1973, Weiss (1973) determined all edge-primitive graphs of valency three, and recently Guo et al. (2013,2015) classified edge-primitive graphs of valencies four and five. In this paper, we determine all edge-primitive Cayley graphs on abelian groups and dihedral groups.     

The traveling salesman problem in graphs with some excluded minors

Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.     

Classifying cubic symmetric graphs of order 10p or 10p2

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ⩾ 1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s-regular cubic graphs of order 10p or 10p ² are also classified for each s ⩾ 1 and each prime p, of which the proof depends on the classification of finite simple groups.     

Minor‐order obstructions for the graphs of vertex cover 6

We provide for the first time, a complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6. This study shows how to limit both the search space of graphs and improve the efficiency of an obstruction checking algorithm when restricted to k–VERTEX COVER graph families. In particular, our upper bounds 2k + 1 (2k + 2) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all k > 0. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 163–178, 2002     

Excluded checkerboard colourable ribbon graph minors

Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.     

Normal cirulant graphs with noncyclic regular subroups

We prove that any circulant graph of order n with connection set S such that n and the order of ?(S), the subgroup of ? that fixes S set‐wise, are relatively prime, is also a Cayley graph on some noncyclic group, and shows that the converse does not hold in general. In the special case of normal circulants whose order is not divisible by 4, we classify all such graphs that are also Cayley graphs of a noncyclic group, and show that the noncyclic group must be metacyclic, generated by two cyclic groups whose orders are relatively prime. We construct an infinite family of normal circulants whose order is divisible by 4 that are also normal Cayley graphs on dihedral and noncyclic abelian groups. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Unavoidable topological minors of infinite graphs

A graph G is loosely-c-connected, or ?-c-connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of ?-c-connected infinite graphs that form an unavoidable set among the topological minors of ?-c-connected infinite graphs. Corresponding results for minors and parallel minors are also obtained.     

Dominating and large induced trees in regular graphs

Recently Chen et al. [Tree domination in graphs, Ars Combin. 73 (2004) 193-203] asked for characterizations of the class of graphs and the class of regular graphs that have an induced dominating tree, i.e. for which there exists a dominating set that induces a tree.We give a somewhat negative answer to their question by proving that the corresponding decision problems are NP-complete. Furthermore, we prove essentially best-possible lower bounds on the maximum order of induced trees in connected cacti of maximum degree 3 and connected cubic graphs.Finally, we give a forbidden induced subgraph condition for the existence of induced dominating trees.     

A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs

Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this article, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position random graphs. We formulate a notion of consistency in this context and propose a valid test for the hypothesis that two finite-dimensional random dot product graphs on a common vertex set have the same generating latent positions or have generating latent positions that are scaled or diagonal transformations of one another. Our test statistic is a function of a spectral decomposition of the adjacency matrix for each graph and our test procedure is consistent across a broad range of alternatives. We apply our test procedure to real biological data: in a test-retest dataset of neural connectome graphs, we are able to distinguish between scans from different subjects; and in the C. elegans connectome, we are able to distinguish between chemical and electrical networks. The latter example is a concrete demonstration that our test can have power even for small-sample sizes. We conclude by discussing the relationship between our test procedure and generalized likelihood ratio tests. Supplementary materials for this article are available online.     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Achievable sets, brambles, and sparse treewidth obstructions

One consequence of the graph minor theorem is that for every k there exists a finite obstruction set Obs(TW?k). However, relatively little is known about these sets, and very few general obstructions are known. The ones that are known are the cliques, and graphs which are formed by removing a few edges from a clique. This paper gives several general constructions of minimal forbidden minors which are sparse in the sense that the ratio of the treewidth to the number of vertices n does not approach 1 as n approaches infinity. We accomplish this by a novel combination of using brambles to provide lower bounds and achievable sets to demonstrate upper bounds. Additionally, we determine the exact treewidth of other basic graph constructions which are not minimal forbidden minors.