An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime (provided solvable vertex stabilizers) are alternating An. This naturally motivates study of for arbitrary valency. In this paper, we characterize automorphism such graphs. particular, show a non-complete (G,2)-arc-transitive graph G simple, socle...

Bandelt and Mulder’s structural characterization of Bipartite Distance Hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pending vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such gra...

We consider the following vertex-partition problem on graphs, known as the CLUSTER DELETION (CD) problem: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case, cliques) to minimize the total weight of edges outside the clusters. The decision version of this optimization problem is known to be NP-complete even for unweighted graphs and has been stu...

In this paper we deal with the arc ranking problem of directed graphs. We give some classes of graphs for which the arc ranking problem is polynomially solvable. We prove that deciding whether χr(G) ≤ 6, where G is an acyclic orientation of a 3-partite graph is an NP-complete problem. In this way we answer an open question stated by Kratochvil and Tuza in 1999.

We use throughout the notation of [5], to which we refer for the basic theory of finite rank 3 permutation groups G. The solvable primitive rank 3 permutation groups have been determined by Foulser [4] and, independently, by Dornhoff [2]. Since rank 3 groups of odd order are solvable we assume that G has even order. Then the graphs JA and JT associated with the non-trivial orbitals A and T of G...

In this paper we study the generalized vertex cover problem (GVC), which is a generalization of various well studied combinatorial optimization problems. GVC is shown to be equivalent to the unconstrained binary quadratic programming problem and also equivalent to some other variations of the general GVC. Some solvable cases are identified and approximation algorithms are suggested for special ...

In this paper we describe properties of a particular transformation of the simple plant location problem into a vertex packing problem on a corresponding undirected graph. We characterize the undirected graphs that arise from simple plant location problems and we give a necessary and su cient condition for such graphs to be perfect. This allows us to nd a new family of polynomially solvable ins...

In this talk, we introduce the character degree graph of a finite group G. We will discuss various properties of this graph, and we will further discuss what graphs can occur as the character degree graphs of finite solvable groups. In particular, we focus on groups with characters of degree pq, pr, and qr, where p, q, and r are distinct primes. Although some group theory will be assumed, knowl...

Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, inclu...

We consider the class C of graphs whose minimal separators have a fixed bounded size. We give an O(nm)-time algorithm computing an optimal tree-decomposition of every graph in C with n vertices and m edges. Furthermore we make evident that many NP-complete problems are solvable in polynomial time when restricted to this class. Both claims hold although C contains graphs of arbitrarily large tre...