Aurora Morgana

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.

A linear algorithm for 2-bend embeddings of planar graphs in the two-dimensional grid

Abstract In this paper we describe a linear algorithm for embedding planar graphs in the rectilinear two-dimensional grid, where vertices are grid points and edges are noncrossing grid paths. The main feature of our algorithm is that each edge is guaranteed to have at most 2 bends (with the single exception of the octahedron for which 3 bends are needed). The total number of bends is at most2 n + 4 if the graph is biconnected and at most(7/3) n in the general case. The area is( n + 1) 2 in the worst case. This problem has several applications to VLSI circuit design, aesthetic layout of diagrams, computational geometry.

General theoretical results on rectilinear embedability of graphs

In the design of certain kinds of electronic circuits the following question arises: given a non-negative integerk, what graphs admit of a plane embedding such that every edge is a broken line formed by horizontal and vertical segments and having at mostk bends? Any such graph is said to bek-rectilinear. No matter whatk is, an obvious necessary condition fork-rectilinearity is that the degree of each vertex does not exceed four.Our main result is that every planar graphH satisfying this condition is 3-rectilinear: in fact, it is 2-rectilinear with the only exception of the octahedron. We also outline a polynomial-time algorithm which actually constructs a plane embedding ofH with at most 2 bends (3 bends ifH is the octahedron) on each edge. The resulting embedding has the property that the total number of bends does not exceed 2n, wheren is the number of vertices ofH.

Degree sequences of matrogenic graphs

Abstract A structural description and a recognition algorithm for matrogenic graphs [4] are given. In fact, matrogenicity is seen to depend only on the degree sequence. An explicit characterization of the degree sequences of matrogenic graphs, and more generally of box-threshold graphs [10], is provided.

On the complexity of recognizing tough graphs

Abstract We consider the relationship between the minimum degree δ of a graph and the complexity of recognizing if a graph is t -tough. Let t ⩾1 be a rational number. We first show that if δ ( G )⩾ tn /( t +1), then G is t -tough. On the other hand, for any fixed e>0, we show that it is NP-hard to determine if G is t -tough, even for the class of graphs with δ ( G )⩾( t /( t +1)− e ) n . In particular, for any fixed c G with δ ( G )⩾ cn .

The clique operator on graphs with few P 4 's

The clique graph of a graph G is the intersection graph K (G) of the (maximal) cliques of G. The iterated clique graphs Kn (G) are defined by KO(G) = G and Ki(G) = K(Ki-1(G)), i > 0 and K is the clique operator. In this article we use the modular decomposition technique to characterize the K-behaviour of some classes of graphs with few P4s.These characterizations lead to polynomial time algorithms for deciding the K-convergence or K-divergence of any graph in the class.

A simple proof of a theorem of Jung

Abstract Jungs theorem states that if G is a 1-tough graph on n ⩾11 vertices such that d ( x ) + d ( y )⩾ n −4 for all distinct nonadjacent vertices x , y , then G is hamiltonian. We give a simple proof of this theorem for graphs with 16 or more vertices.

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 complexity of recognizing tough cubic graphs

We show that it is NP-hard to determine if a cubic graph G is 1-tough. We then use this result to show that for any integer t 1, it is NP-hard to determine if a 3t-regular graph is t-tough. We conclude with some remarks concerning the complexity of recognizing certain subclasses of tough graphs.

An algorithm for 1-bend embeddings of plane graphs in the two-dimensional grid

In this paper we characterize the class of plane graphs that can be embedded on the two-dimensional grid with at most one bend on each edge. In addition, we provide an algorithm that either detects a forbidden configuration or generates an embedding with at most one bend on each edge.