On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study ‘almost distance-regular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity or, informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (?,m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.     

Circulant graph imbeddings

An algebraic characterization is given for those Cayley graphs for cyclic groups in which the neighborhood of any vertex is a cycle. A triangular imbedding is obtained for each such graph, either in the sphere, the torus, or the Klein bottle. The automorphism groups of these graphs are determined, and necessary and sufficient conditions are given for two such graphs to be isomorphic.     

Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

A two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V → ℝ². We view (G,p) as a straight line realization of G in ℝ². Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].     

On transitive one-factorizations of arc-transitive graphs

It is shown that transitive 1-factorizations of arc-transitive graphs exist if and only if certain factorizations of their automorphism groups exist. This relation provides a method for constructing and characterizing transitive 1-factorizations for certain classes of arc-transitive graphs. Then a characterization is given of 2-arc-transitive graphs which have transitive 1-factorizations. In this characterization, some 2-arc transitive graphs and their transitive 1-factorizations are constructed.     

A separation property of planar triangulations

Every planar triangulation G has the property that each induced cycle C of length at least 4 in G separates G, but no proper subgraph of C does. This property is trivially shared by all chordal graphs since these contain no such cycles at all. We ask to what extent maximally planar graphs and chordal graphs are unique with this property — or how much larger the class of graphs is that it determines. The answer is given in the form of a characterization of this class in terms of the simplicial decompositions of its elements. The theory of simplicial decompositions appears to be a very interesting, but still largely unexploited, method of characterization in graph theory, which seems tailor-made for problems like the one discussed.     

Greedily constructing Hamiltonian paths, Hamiltonian cycles and maximum linear forests

There are various greedy schemas to construct a maximal path in a given input graph. Associated with each such schema is the family of graphs where it always results a path of maximum length, or a Hamiltonian path/cycle, if there exists one. Considerable amount of work has been carried out, regarding the characterization and recognition problems of such graphs. We present here a systematic listing of previous results of this type and fill up most of the remaining empty entries.     

Weakly bipartite graphs and the Max-cut problem

A new class of graphs, called weakly bipartite graphs, is introduced. A graph is called weakly bipartite if its bipartite subgraph polytope coincides with a certain polyhedron related to odd cycle constraints. The class of weakly bipartite graphs contains for instance the class of bipartite graphs and the class of planar graphs. It is shown that the max-cut problem can be solved in polynomial time for weakly bipartite graphs. The polynomical algorithm presented is based on the ellipsoid method and an algorithm that computes a shortest path of even length.     

A graphical characterization of the largest chain graphs

The paper presents a graphical characterization of the largest chain graphs which serve as unique representatives of classes of Markov equivalent chain graphs. The characterization is a basis for an algorithm constructing, for a given chain graph, the largest chain graph equivalent to it. The algorithm was used to generate a catalog of the largest chain graphs with at most five vertices. Every item of the catalog contains the largest chain graph of a class of Markov equivalent chain graphs and an economical record of the induced independency model.     

On the vertex packing problem

A forbidden subgraphs characterization of the class of graphs that arise from bipartite graphs, odd holes, and graphs with no complement of a diamond via repeated substitutions is given. This characterization allows us to solve the vertex packing problem for the graphs in this class.     

Claw-free graphs with strongly perfect complements. Fractional and integral version, Part II: Nontrivial strip-structures

Strongly perfect graphs have been studied by several authors (e.g., Berge and Duchet (1984) [1], Ravindra (1984) [7] and Wang (2006) [8]). In a series of two papers, the current paper being the second one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [8] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.     

Claw-free graphs with strongly perfect complements. Fractional and integral version. Part I. Basic graphs

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [14] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.     

A class of h-perfect graphs

A graph is said to be h-perfect if the convex hull of its independent sets is defined by the constraints corresponding to cliques and odd holes, and the nonnegativity constraints. Series-parallel graphs and perfect graphs are h-perfect. The purpose of this paper is to extend the class of graphs known to be h-perfect. Thus, given a graph which is the union of a bipartite graph G₁ and a graph G₂ having exactly two common nodes a and b, and no edge in common, we prove that G is h-perfect if so is the graph obtained from G by replacing G₁ by an a-b chain (the length of which depends on G₁). This result enables us to prove that the graph obtained by substituting bipartite graphs for edges of a series-parallel graph is h-perfect, and also that the identification of two nodes of a bipartite graph yields an h-perfect graph (modulo a reduction which preserves h-perfection).     

Scheduling unit-length jobs with precedence constraints of small height

We consider the problem of scheduling unit-length jobs on identical machines subject to precedence constraints. We show that natural scheduling rules fail when the precedence constraints form a collection of stars or a collection of complete bipartite graphs. We prove that the problem is in fact NP-hard on collections of stars when the input is given in a compact encoding, whereas it can be solved in polynomial time with standard adjacency list encoding. On a subclass of collections of stars and on collections of complete bipartite graphs we show that the problem can be solved in polynomial time even when the input is given in compact encoding, in both cases via non-trivial algorithms.     

Tree loop graphs

Many problems involving DNA can be modeled by families of intervals. However, traditional interval graphs do not take into account the repeat structure of a DNA molecule. In the simplest case, one repeat with two copies, the underlying line can be seen as folded into a loop. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and, in the affirmative case, produces a loop representation for the tree.     

On metric properties of certain clique graphs

The paper provides a unified point of view on some classes of graphs: clique graphs, weakly geodetic graphs, ptolemaic graphs and Husimi trees. A purely metric characterization of Husimi trees is given.     

Chordal graphs and join spaces

A join space is an abstract model for partially ordered linear, spherical and projective geometries. A characterization for chordal graphs which are join spaces is given: a chordal graph is a join space if and only if it does not contain one of the two forbidden graphs as an induced subgraph.     

The Laplacian spectral radius for unicyclic graphs with given independence number

In this paper, we give a complete characterization of the extremal graphs with maximal Laplacian spectral radius among all unicyclic graphs with given order and given number of pendent vertices. Then we study the Laplacian spectral radius of unicyclic graphs with given independence number and characterize the extremal graphs completely.     

On the spectral characterization of the union of complete multipartite graph and some isolated vertices

A graph is said to be determined by its adjacency spectrum (DS for short) if there is no other non-isomorphic graph with the same spectrum. In this paper, we focus our attention on the spectral characterization of the union of complete multipartite graph and some isolated vertices, and all its cospectral graphs are obtained. By the results, some complete multipartite graphs determined by their adjacency spectrum are also given. This extends several previous results of spectral characterization related to the complete multipartite graphs.     

On Graphs Without Multicliqual Edges

An edge which belongs to more than one clique of a given graph is called a multicliqual edge. We find a necessary and sufficient condition for a graph H to be the clique graph of some graph G without multicliqual edges. We also give a characterization of graphs without multicliqual edges that have a unique critical generator. Finally, it is shown that there are infinitely many self-clique graphs having more than one critical generator.