Douglas Bauer

Domination alteration sets in graphs

The domination number @a(G) of a graph G is the size of a minimum dominating set, i.e., a set of points with the property that every other point is adjacent to a point of the set. In general @a(G) can be made to increase or decrease by the removal of points from G. Our main objective is the study of this phenomenon. For example we show that if T is a tree with at least three points then @a(T - v) > @a (T) if and only if @n is in every minimum dominating set of T. Removal of a set of lines from a graph G cannot decrease the domination number. We obtain some upper bounds on the size of a minimum set of lines which when removed from G increases the domination number.

Recognizing tough graphs is NP-hard

We show that recognizing 1-tough graphs is NP-hard, thereby settling a long-standing open problem. We also prove it is NP-hard to recognize t-tough graphs for any fixed positive rational number t.

Not every 2-tough graph is Hamiltonian

We present (9/4-e)-tough graphs without a Hamilton path for arbitrary >0, thereby refuting a well-known conjecture due to Chvatal. We also present (7/4-e) -tough chordal graphs without a Hamilton path for any e>0.

Combinatorial optimization problems in the analysis and design of probabilistic networks

This paper presents some results regarding the design of reliable networks. The problem under consideration involves networks which are undirected graphs having equal and independent edge failure probabilities. The index of reliability is the probability that the network fails (becomes disconnected). For “small” edge failure probabilities and given p and q there exists a class of p vertex, q edge graphs with the property that any graph in the class has a smaller probability of disconnection than any graph outside of the class. We solve the problem of synthesizing graphs in this class.

Toughness in graphs - A survey

In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!

Long cycles in graphs with large degree sums

Abstract A number of results are established concerning long cycles in graphs with large degree sums. Let G be a graph on n vertices such that d(x)+d(y)+d(z)⩾s for all triples of independent vertices x, y, z. Let c be the length of a longest cycle in G and α the cardinality of a maximum independent set of vertices. If G is 1-tough and s⩾n, then every longest cycle in G is a dominating cycle and c⩾ min (n, n+ 1 3 s−α)⩾ min (n, 1 2 n+ 1 3 s)⩾ 5 6 n . If G is 2-connected and s⩾n+2, then also c⩾ min (n, n+ 1 3 s-α) , generalizing a result of Bondy and one of Nash-Williams. Finally, if G is 2-tough and s⩾n, then G is hamiltonian.

Hamiltonian properties of graphs with large neighborhood unions

Abstract Let G be a graph of order n , σ k = min{ ϵ i =1 k d ( ν i ): { ν 1 ,…, ν k } is an independent set of vertices in G }, NC = min{| N ( u )∪ N ( ν )|: uν ∉ E ( G )} and NC2 = min{| N ( u )∪ N ( ν )|: d ( u , ν )=2}. Ore proved that G is hamiltonian if σ 2 ⩾ n ⩾3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and NC ⩾ 1 3 (2n−1) . It is shown that both results are generalized by a recent result of Bauer et al. Various other existing results in hamiltonian graph theory involving degree-sums or cardinalities of neighborhood unions are also compared in terms of generality. Furthermore, some new results are proved. In particular, it is shown that the bound 1 3 (2n−1) on NC in the result of Faudree et al. can be lowered to 1 3 (2n−1) , which is best possible. Also, G is shown to have a cycle of length at least min{ n , 2(NC2)} if G is 2-connected and σ 3 ⩾ n +2. A D λ -cycle ( D λ -path) of G is a cycle (path) C such that every component of G − V ( C ) has order smaller than λ. Sufficient conditions of Lindquester for the existence of Hamilton cycles and paths involving NC2 are extended to D λ -cycles and D λ -paths.

Long cycles in graphs with prescribed toughness and minimum degree

A cycle C of a graph G is a Dλ-cycle if every component of G-V(C) has order less than λ. Using the notion of Dλ-cycles, a number of results are established concerning long cycles in graphs with prescribed toughness and minimum degree. Let G be a t-tough graph on n 3 vertices. If δ > n/(t + λ) + λ − 2 for some λ t + 1, then G contains a Dλ-cycle. In particular, if δ > n/(t + 1) − 1, then G is hamiltonian, improving a classical result of Dirac for t> 1. If G is nonhamiltonian and δ > n/(t + λ) + λ − 2 for some λ t + 1, then G contains a cycle of length at least (t + 1)(δ − λ + 2) + t, partially improving another classical result of Dirac for t> 1.

Toughness, minimum degree, and the existence of 2-factors

Degree conditions on the vertices of a t-tough graph G(1 ≦ t ≦ 2) that ensure the existence of a 2-factor in G are presented. These conditions are asymptotically best possible for every t ϵ [1, 3/2] and for infinitely many t ϵ [3/2, 2].

Hamiltonian degree conditions which imply a graph is pancyclic

Abstract We use a recent cycle structure theorem to prove that three well-known hamiltonian degree conditions (due to Chvatal, Fan, and Bondy) each imply that a graph is either pancyclic, bipartite, or a member of an easily identified family of exceptions.