K3,3-free intersection graphs of finite groups

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H∩K≠1 where 1 denotes the trivial subgroup of G. In this paper we classify all finite groups whose intersection graphs are K_3,3-free.     

On the intersection graph of a finite group

For a finite group G, the intersection graph of G which is denoted by Γ(G) is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H ∩ K ≠ 1. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of Aut(Γ(G)).     

On supercompact graphs

A graph G is called a supercompact graph if G is the intersection graph of some family ?? of subsets of a set X such that ?? satisfies the Helly property and for any x≠y in X, there exists S ∈ ?? with x ∈ S, y ? S. Various characterizations of supercompact graphs are given. It is shown that every clique-critical graph is supercompact. Furthermore, for any finite graph, H, there is at most a finite number of different supercompact graphs G such that H is the clique-graph of G.     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

Bounding the Size of Equimatchable Graphs of Fixed Genus

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In an earlier paper with Saito, the authors showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only a finite number of 3-connected equimatchable graphs having a minimal embedding of representativity at least three. (In fact, if the graphs considered are non-bipartite, the representativity three hypothesis may be dropped.) The proof makes use of the Gallai-Edmonds decomposition theorem for matchings.      

Intersection Graphs and the Clique Operator

 Let P be a class of finite families of finite sets that satisfy a property P. We call ΩP the class of intersection graphs of families in P and CliqueP the class of graphs whose family of cliques is in P. We prove that a graph G is in ΩP if and only if there is a family of complete sets of G which covers all edges of G and whose dual family is in P. This result generalizes that of Gavril for circular-arc graphs and conduces those of Fulkerson-Gross, Gavril and Monma-Wei for interval graphs, chordal graphs, UV, DV and RDV graphs. Moreover, it leads to the characterization of Helly-graphs and dually chordal graphs as classes of intersection graphs. We prove that if P is closed under reductions, then CliqueP=Ω(P ^*∩H) (P ^*= Class of dual families of families in P). We find sufficient conditions for the Clique Operator, K, to map ΩP into ΩP ^*. These results generalize several known results for particular classes of intersection graphs. Furthermore, they lead to the Roberts-Spencer characterization for the image of K and the Bandelt-Prisner result on K-fixed classes. Received: August 18, 1997 Final version received: March 30, 1999     

On the computational complexity of ordered subgraph recognition

Let (G, <) be a finite graph G with a linearly ordered vertex set V. We consider the decision problem (G, <)ORD to have as an instance an (unordered) graph Γ and as a question whether there exists a linear order ≤ on V(Γ) and an order preserving graph isomorphism of (G, <) onto an induced subgraph of Γ. Several familiar classes of graph are characterized as the yes-instances of (G, < )ORD for appropriate choices of (G, <). Here the complexity of (G, <)ORD is investigated. We conjecture that for any 2-connected graph G, G ≠ K_k, (G, <)ORD is NP-complete. This is verified for almost all 2-connected graphs. Several related problems are formulated and discussed. © 1995 John Wiley & Sons, Inc.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Decomposition of 3-connected graphs

Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K _3,n for somen3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.     

Partial characterizations of clique-perfect and coordinated graphs: Superclasses of triangle-free graphs

A graph G is clique-perfect if the cardinality of a maximum clique-independent set of H equals the cardinality of a minimum clique-transversal of H, for every induced subgraph H of G. A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem, W₄, bull}-free, both superclasses of triangle-free graphs.     

On possible counterexamples to Negami's planar cover conjecture

A simple graph H is a cover of a graph G if there exists a mapping φ from H onto G such that φ maps the neighbors of every vertex υ in H bijectively to the neighbors of φ (υ) in G . Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, and the first author that the conjecture holds as long as the graph K _1,2,2,2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 183–206, 2004     

F-Continuous Graphs

For a nontrivial connected graph F, the F-degree of a vertex in a graph G is the number of copies of F in G containing . A graph G is F-continuous (or F-degree continuous) if the F-degrees of every two adjacent vertices of G differ by at most 1. All P₃-continuous graphs are determined. It is observed that if G is a nontrivial connected graph that is F-continuous for all nontrivial connected graphs F, then either G is regular or G is a path. In the case of a 2-connected graph F, however, there always exists a regular graph that is not F-continuous. It is also shown that for every graph H and every 2-connected graph F, there exists an F-continuous graph G containing H as an induced subgraph.     

On constructible graphs,locally Helly graphs,and convexity

A (finite or infinite) graph G is constructible if there exists a well‐ordering ≤ of its vertices such that for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x. Particular constructible graphs are Helly graphs and connected bridged graphs. In this paper we study a new class of constructible graphs, the class of locally Helly graphs. A graph G is locally Helly if, for every pair (x,y) of vertices of G whose distance is d ≥ 2, there exists a vertex whose distance to x is d ? 1 and which is adjacent to y and to all neighbors of y whose distance to x is at most d. Helly graphs are locally Helly, and the converse holds for finite graphs. Among different properties we prove that a locally Helly graph is strongly dismantable, hence cop‐win, if and only if it contains no isometric rays. We show that a locally Helly graph G is finitely Helly, that is, every finite family of pairwise non‐disjoint balls of G has a non‐empty intersection. We give a sufficient condition by forbidden subgraphs so that the three concepts of Helly graphs, of locally Helly graphs and of finitely Helly graphs are equivalent. Finally, generalizing different results, in particular those of Bandelt and Chepoi 1 about Helly graphs and bridged graphs, we prove that the Helly number h(G) of the geodesic convexity in a constructible graph G is equal to its clique number ω(G), provided that ω(G) is finite. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 280–298, 2003     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

Minimal extensions of graphs to absolute retracts

A graph H is an absolute retract if for every isometric embedding h of, into a graph G an edge-preserving map g from G to H exists such that g · h is the identity map on H. A vertex v is embeddable in a graph G if G ? v is a retract of G. An absolute retract is uniquely determined by its set of embeddable vertices. We may regard this set as a metric space. We also prove that a graph (finite metric space with integral distance) can be isometrically embedded into only one smallest absolute retract (injective hull). All graphs in this paper are finite, connected, and without multiple edges.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non-bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

A characterization of domination perfect graphs

Let γ(G) and i(G) be the domination number and independent domination number of a graph G, respectively. Sumner and Moore [8] define a graph G to be domination perfect if γ(H) = i(H), for every induced subgraph H of G. In this article, we give a finite forbidden induced subgraph characterization of domination perfect graphs. Bollobás and Cockayne [4] proved an inequality relating γ(G) and i(G) for the class of K^1,k -free graphs. It is shown that the same inequality holds for a wider class of graphs.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.