On vertex k-partitions of certain infinite graphs

Let G be an infinite graph; define deg∞ G to be the least m such that any partition P of the vertex set of G into sets of uniformly bounded cardinality contains a set which is adjacent to at least m other sets of the partition. If G is either a regular tree on a triangular, square or hexagonal planar mosaic graph, it is shown that deg∞ G equals the degree of G. This verifies some conjectures of S. Ulam. Several open problems are given.     

Tree spanners in planar graphs

A tree t-spanner of a graph G is a spanning subtree T of G in which the distance between every pair of vertices is at most t times their distance in G. Spanner problems have received some attention, mostly in the context of communication networks. It is known that for general unweighted graphs, the problem of deciding the existence of a tree t-spanner can be solved in polynomial time for t=2, while it is NP-hard for any t⩾4; the case t=3 is open, but has been conjectured to be hard. In this paper, we consider tree spanners in planar graphs. We show that even for planar unweighted graphs, it is NP-hard to determine the minimum t for which a tree t-spanner exists. On the other hand, we give a polynomial algorithm for any fixed t that decides for planar unweighted graphs with bounded face length whether there is a tree t-spanner. Furthermore, we prove that it can be decided in polynomial time whether a planar unweighted graph has a tree t-spanner for t=3.     

A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X|≥3 and if X is 4-connected in G. If X=V(G) then Sanders’ result follows.     

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

The number of edge 3-colorings of a planar cubic graph as a permanent

In this paper, we establish that the number of edge 3-colorings of a finite planar cubic graph G, i.e., 3-colorings of its interchange graph H, is equal to 2^?N|Permanent(A)|, where N is the number of edges of G, and A is the 2N × 2N square matrix formed by repeating each row of the N × 2N vertex-directed edge incidence matrix of H (with an arbitrary orientation). As the number of 4-colorings of a finite maximal (fully triangulated) planar graph is equal to four times the number of edge 3-colorings of its planar cubic dual, this result gives a formula for the number of 4-colorings of a finite maximal planar graph.     

On the vertex face total chromatic number of planar graphs

Let G be a planar graph. The vertex face total chromatic number _χ13(G) of G is the least number of colors assigned to V(G)∪F(G) such that no adjacent or incident elements receive the same color. The main results of this paper are as follows: (1) We give the vertex face total chromatic number for all outerplanar graphs and modulus 3-regular maximal planar graphs. (2) We prove that if G is a maximal planar graph or a lower degree planar graph, i.e., ∠(G) ≤ 3, then _χ13(G) ≤ 6. © 1996 John Wiley & Sons, Inc.     

On well-covered triangulations: Part III

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper Finbow et al. (2004) [3] that there are no 5-connected planar well-covered triangulations, and in Finbow et al. (submitted for publication) [4] that there are exactly four 4-connected well-covered triangulations containing two adjacent vertices of degree 4. It is the aim of the present paper to complete the characterization of 4-connected well-covered triangulations by showing that each such graph contains two adjacent vertices of degree 4.     

On a conjecture of Meyniel

A graph G is said to be very strongly perfect if for each induced subgraph H of G, each vertex of H belongs to a stable set that meets all maximal cliques of H. Meyniel proved that a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords. Nowadays, such a graph is called a Meyniel graph. We prove that, as conjectured by Meyniel, a graph is very strongly perfect if and only if it is a Meyniel graph. We also design a polynomial-time algorithm which, given a Meyniel graph G and a vertex x of G, finds a stable set that contains x and meets all maximal cliques of G. We shall convert this algorithm into another polynomial-time algorithm which, given a Meyniel graph G, finds an optimal coloring of G, and a largest clique of G. Finally, we shall establish another property, related to perfection, of Meyniel graphs.     

Enumeration aspects of maximal cliques and bicliques

We present a general framework to study enumeration algorithms for maximal cliques and maximal bicliques of a graph. Given a graph G, we introduce the notion of the transition graph T(G) whose vertices are maximal cliques of G and arcs are transitions between cliques. We show that T(G) is a strongly connected graph and characterize a rooted cover tree of T(G) which appears implicitly in [D.S. Johnson, M. Yannakakis, C.H. Papadimitriou, On generating all maximal independent sets, Information Processing Letters 27 (1988) 119-123; S. Tsukiyama, M. Ide, M. Aiyoshi, I. Shirawaka, A new algorithm for generating all the independent sets, SIAM Journal on Computing 6 (1977) 505-517]. When G is a bipartite graph, we show that the Galois lattice of G is a partial graph of T(G) and we deduce that algorithms based on the Galois lattice are a particular search of T(G). Moreover, we show that algorithms in [G. Alexe, S. Alexe, Y. Crama, S. Foldes, P.L. Hammer, B. Simeone, Consensus algorithms for the generation of all maximal bicliques, Discrete Applied Mathematics 145 (1) (2004) 11-21; L. Nourine, O. Raynaud, A fast algorithm for building lattices, Information Processing Letters 71 (1999) 199-204] generate maximal bicliques of a bipartite graph in O(n²) per maximal biclique, where n is the number of vertices in G. Finally, we show that under some specific numbering, the transition graph T(G) has a hamiltonian path for chordal and comparability graphs.     

Graphs of given genus and arbitrarily large maximum genus

The maximum genus of a connected graph G is the maximum among the genera of all compact orientable 2-manifolds upon which G has 2-cell embeddings. In the theorems that follow the use of an edge-adding technique is combined with the well-known Edmonds' technique to produce the desired results. Planar graphs of arbitrarily large maximum genus are displayed in Theorem 1. Theorem 2 shows that the possibility for arbitrarily large difference between genus and maximum genus is not limited to planar graphs. In particular, we show that the wheel graph, the standard maximal planar graph, and the prism graph are upper embeddable. We then show that given m and n, there is a graph of genus n and maximum genus larger than mn.     

Connectivity and tree structure in finite graphs

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditionsunder which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the k-blocks — the maximal vertex sets that cannot be separated by at most k vertices — of a graph G live in distinct parts of a suitable treedecomposition of G of adhesion at most k, whose decomposition tree is invariant under the automorphisms of G. This extends recent work of Dunwoody and Krön and, like theirs, generalizes a similar theorem of Tutte for k=2. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all k simultaneously, all the k-blocks of a finite graph.     

Representing edge intersection graphs of paths on degree 4 trees

Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P, denoted by EPT(P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G=EPT(P) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph.An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.     

On maximum planar induced subgraphs

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G^′. In this case G^′ is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G)?k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G)?k. We prove that unless P=NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.     

Subgraphs with Restricted Degrees of Their Vertices in Planar 3-Connected Graphs

We have proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path on k vertices; the bound 5k is the best possible. Moreover, for every connected planar graph H other than a path and for every integer m ≥ 3 there is a 3-connected planar graph G such that each copy of H in G contains a vertex of degree at least m.     

Subclasses of k-trees: Characterization and recognition

A k-tree is either a complete graph on k vertices or a graph G=(V,E) that contains a vertex whose neighbourhood in G induces a complete graph on k vertices and whose removal results in a k-tree. We present two new subclasses of k-trees and their properties. First, we present the definition and characterization of k-path graphs, based on the concept of k-paths, that generalizes the classic concept of paths. We also introduce the simple-clique k-trees, of which the maximal outerplanar graphs and the planar 3-trees are particular cases. Based on Characterization Theorems, we show recognition algorithms for both families. Finally, we establish the inclusion relations among these new classes and k-trees.     

A vertex incremental approach for maintaining chordality

For a chordal graph G=(V,E), we study the problem of whether a new vertex u∉V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

Destroying automorphisms by fixing nodes

The fixing number of a graph G is the minimum cardinality of a set S⊂V(G) such that every nonidentity automorphism of G moves at least one member of S, i.e., the automorphism group of the graph obtained from G by fixing every node in S is trivial. We provide a formula for the fixing number of a disconnected graph in terms of the fixing numbers of its components and make some observations about graphs with small fixing numbers. We determine the fixing number of a tree and find a necessary and sufficient condition for a tree to have fixing number 1.     

The intersection graphs of subtrees in trees are exactly the chordal graphs

The intersection graph of a family of subtrees in an undirected tree is called a subtree graph. A graph is called chordal if every simple circuit with more than three vertices has an edge connecting two non-consecutive vertices. In this paper, we prove that, for a graph G, the following conditions are equivalent: (i) G is a chordal graph; (ii) G is a subtree graph; (iii) G is a proper subtree graph.Consider a chordal graph G. We give an efficient algorithm for constructing a representation of G by a family of subtrees in a tree.     

Planar graphs without 3-, 7-, and 8-cycles are 3-choosable

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem [4] states that every planar triangle-free graph is 3-colorable. However, Voigt [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangle-free graph without 7- and 8-cycles is 3-choosable.