The Isomorphism Problem For Directed Path Graphs and For Rooted Directed Path Graphs

This paper deals with the isomorphism problem of directed path graphs and rooted directed path graphs. Both graph classes belong to the class of chordal graphs, and for both classes the relative complexity of the isomorphism problem is yet unknown. We prove that deciding isomorphism of directed path graphs is isomorphism complete, whereas for rooted directed path graphs we present a polynomial-time isomorphism algorithm.     

Graphs with two isomorphism classes of spanning unicyclic subgraphs

The graphs whose spanning unicyclic subgraphs partition into exactly two isomorphism classes are characterized.This work is a continuation of [6] where graphs with one isomorphism class of spanning unicyclic graphs are characterized. The analogous question for spanning trees was posed in [10] and graphs with one isomorphism class of spanning trees were characterized in [2], [3], [4], [7], [11] while graphs with two isomorphism classes of spanning trees were characterized in [4], [5]. Related topics are treated in [1], [8], [9].     

Even signings,signed switching classes,and (−1, 1)-matrices

In this paper, algebraic and combinatorial techniques are used to establish results concerning even signings of graphs, switching classes of signed graphs, and (?1, 1)-matrices. These results primarily deal with enumeration of isomorphism types, and determining whether there are fixed elements under the action of automorphisms. A formula is given for the number of isomorphism types of even signings of any fixed simple graph. This is shown to be equal to the number of isomorphism types of switching classes of signings of the graph. A necessary and sufficient criterion is found for all switching classes fixed by a given graph automorphism to contain signings fixed by that automorphism. It is determined whether this criterion is met for all automorphisms of various graphs, including complete graphs, which yields a known result of Mallows and Sloane. As an application, a formula is developed for the number of H-equivalence classes of (?1, 1)-matrices of fixed size. Independently, using Molien's theorem and following a suggestion of Cameron's, generating series for these numbers are given. As a final application, a necessary and sufficient condition that a square (?1, 1)-matrix be switching equivalent to a symmetric matrix is given.     

Representative graphs of r-regular partial planes and representation of orthomodular posets

In this paper the authors establish a one-to-one correspondence between the isomorphism classes of graphs whose structure spaces are r-regular partial planes and the isomorphism classes of graphs which have no induced subgraphs isomorphic to an (r + 1)-pointed star or to K₄ with a single line removed. Moreover the graphs whose structure space contains no 4-loops correspond to graphs with no 4-cycles as induced subgraphs. It is pointed out that these graphs give rise to orthomodular posets (or lattices in the case of no 4-loops) and hence are of much interest in the study of orthomodular structures and in particular of quantum logics.     

Definability in the Substructure Ordering of Simple Graphs

For simple graphs, we investigate and seek to characterize the properties first-order definable by the induced subgraph relation. Let \({\mathcal{P}\mathcal{G}}\) denote the set of finite isomorphism types of simple graphs ordered by the induced subgraph relation. We prove this poset has only one non-identity automorphism co, and for each finite isomorphism type G, the set {G, G ^co } is definable. Furthermore, we show first-order definability in \({\mathcal{P}\mathcal{G}}\) captures, up to isomorphism, full second-order satisfiability among finite simple graphs. These results can be utilized to explore first-order definability in the closely associated lattice of universal classes. We show that for simple graphs, the lattice of universal classes has only one non-trivial automorphism, the set of finitely generated and finitely axiomatizable universal classes are separately definable, and each such universal subclass is definable up to the unique non-trivial automorphism.     

Fast generation of regular graphs and construction of cages

The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. In this article an efficient algorithm to generate regular graphs with a given number of vertices and vertex degree is introduced. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. The implementation allows computing even large classes of graphs, like construction of the 4‐regular graphs on 18 vertices and, for the first time, the 5‐regular graphs on 16 vertices. Also in cases with given girth, some remarkable results are obtained. For instance, the 5‐regular graphs with girth 5 and minimal number of vertices were generated in less than 1 h. There exist exactly four (5, 5)‐cages. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 137–146, 1999     

Locally petersen graphs

A graph Γ is locally Petersen if, for each point t of Γ, the graph induced by Γ on all points adjacent to t is isomorphic to the Petersen graph. We prove that there are exactly three isomorphism classes of connected, locally Petersen graphs and further characterize these graphs by certain of their parameters.     

Graph isomorphism and equality of simplices

We show that the graph isomorphism problem is equivalent to the problem of recognizing equal simplices in ? ⁿ . This result can lead to new methods in the graph isomorphism problem based on geometrical properties of simplices. In particular, relations between several well-known classes of invariants of graphs and geometrical invariants of simplices are established.     

On the power graph and the reduced power graph of a finite group

In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we investigate the determination of the orders of the elements of a finite group from its directed (resp. undirected) reduced power graph. Consequently, we show that some classes of finite groups are recognizable from their undirected reduced power graphs. Also, we study the relationship between the isomorphism classes of groups corresponding to the equivalence relations induced by the isomorphism of each of these graphs on the set of all finite groups.     

Enumeration of weak isomorphism classes of signed graphs

A signed graph is a graph in which each line has a plus or minus sign. Two signed graphs are said to be weakly isomorphic if their underlying graphs are isomorphic through a mapping under which signs of cycles are preserved, the sign of a cycle being the product of the signs of its lines. Some enumeration problems implied by such a definition, including the problem of self-dual configurations, are solved here for complete signed graphs by methods of linear algebra over the two-element field. It is also shown that weak isomorphism classes of complete signed graphs are equal in number to other configurations: unlabeled even graphs, two-graphs and switching classes.     

Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs

Recursive procedures are obtained for counting isomorphism classes of three-connected graphs and of two-connected graphs without vertices of degree 2. We apply an enumeration tool developed by R. W. Robinson to count non-isomorphic 2-connected graphs: he expressed the sums of cycle indices of automorphism groups of connected graphs in terms of those of their 2-connected components, and we do the same for 2-connected graphs and their 3-connected components.     

Some classical combinatorial problems on circulant and claw-free graphs: the isomorphism and coloring problems on circulant graphs and the stable set problem on claw-free graphs

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Università di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.     

Arithmetical properties of finite graphs and polynomials

A study is made of asymptotic arithmetical properties of the isomorphism classes of certain types of finite graphs, and of certain polynomials over Galois fields.     

Some Families of Directed Strongly Regular Graphs Obtained from Certain Finite Incidence Structures

We present many new directed strongly regular graphs by direct construction. We construct these graphs on the collections of antiflags of certain finite incidence structures. In this way, we confirm the feasibility of infinitely many parameter sets that was previously undetermined. We describe some examples of graphs together with their isomorphism classes to demonstrate the fact that our construction method is capable of producing many graphs with same parameters.     

Special Classes of Snarks

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.     

Fraternal Augmentations of graphs,Coloration and Minors

We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ${\mathrm{\nabla }}_r(G)$. We generalize to these classes some results proved for proper minor closed classes and bounded degree graphs, such as the existence of low tree-width colorings, a linear time algorithm to check subgraph isomorphism for a fixed pattern and homomorphism dualities.     

Natural Isomorphism Classes of Cycle Permutation Graphs Over a Graph

 A permutation graph over a graph was introduced by Lee and Sohn in [8] as a generalization of both a graph bundle over a graph and a standard permutation graph, and they gave a characterization of a natural isomorphism and an automorphism of a given permutation graph over a graph. In this paper, we enumerate the natural isomorphism classes of cycle permutation graphs over a graph. Received: October 21, 1996 Revised: August 25, 1997     

Graphs realised by r.e. equivalence relations

We investigate dependence of recursively enumerable graphs on the equality relation given by a specific r.e. equivalence relation on ω. In particular we compare r.e. equivalence relations in terms of graphs they permit to represent. This defines partially ordered sets that depend on classes of graphs under consideration. We investigate some algebraic properties of these partially ordered sets. For instance, we show that some of these partial ordered sets possess atoms, minimal and maximal elements. We also fully describe the isomorphism types of some of these partial orders.     

Polynomial isomorphism algorithm for graphs which do not pinch to K3,g

In this paper we construct a polynomial algorithm for verifying the isomorphism of graphs which do not pinch to K_3,g. In the construction we use properties of colored graphs from this class and results of Babai-Luks on canonization of graphs. The class cited contains the class of graphs of genus not exceeding g, and we apply the algorithm constructed to the recognition of isomorphism of graphs of bounded valence. The method given is a generalization of the combinatorial and group-theoretic approach to the isomorphism problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 99–114, 1984.