Márcia R. Cerioli

The homogeneous set sandwich problem

Abstract The graph sandwich problem for property Φ is defined as follows: Given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) such that E 1 ⊆ E 2 , is there a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 which satisfies property Φ? We present a polynomialtime algorithm for solving the graph sandwich problem, when property Φ is “to contain a homogeneous set”. The algorithm presented also provides the graph G and a homogeneous set in G in case it exists.

On minimum clique partition and maximum independent set on unit disk graphs and penny graphs: complexity and approximation

Abstract A graph G is a unit disk graph if it is the intersection graph of a family of unit disks in the euclidean plane. If the disks do not overlap, then G is also a unit coin graph or penny graph. In this work we establish the complexity of the minimum clique partition problem and the maximum independent set problem for penny graphs, both NP-complete, and present two approximation algorithms for finding clique partitions: a 3-approximation algorithm for unit disk graphs and a 3 2 -approximation algorithm for penny graphs.

Clique-Coloring Circular-Arc Graphs

Abstract A clique-coloring of a graph is a coloring of its vertices such that no maximal clique of size at least two is monochromatic. A circular-arc graph is the intersection graph of a family of arcs in a circle. We show that every circular-arc graph is 3-clique-colorable. Moreover, we characterize which circular-arc graphs are 2-clique-colorable. Our proof is constructive and gives a polynomial-time algorithm to find an optimal clique-coloring of a given circular-arc graph.

On L(2,1)-coloring split, chordal bipartite, and weakly chordal graphs

An L(2,1)-coloring, or @l-coloring, of a graph is an assignment of non-negative integers to its vertices such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. Given a graph G, @l is the minimum range of colors for which there exists a @l-coloring of G. A conjecture by Griggs and Yeh [J.R. Griggs, R.K. Yeh, Labelling graphs with a condition at distance 2, SIAM Journal on Discrete Mathematics 5 (1992) 586-595] states that @l is at most @D^2, where @D is the maximum degree of a vertex in G. We prove that this conjecture holds for weakly chordal graphs. Furthermore, we improve the known upper bounds for chordal bipartite graphs, and for split graphs.

Edge clique graphs and some classes of chordal graphs

The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge clique graph is that the sizes of all maximal cliques and intersections of maximal cliques ought to be triangular numbers. We show that this condition is also sufficient for starlike-threshold graphs.

On counting interval lengths of interval graphs

Given an interval graph G, the interval count problem is that of computing the minimum number IC(G) of interval lengths needed to represent G. Although the problem of deciding whether IC(G)=1 is equivalent to that of recognizing unit-interval graphs, which is a well-known problem having several efficient recognition approaches, very little is known about deciding efficiently whether IC(G)=k for fixed k>=2. We provide efficient computations of the interval count of generalizations of threshold graphs.

Clique Graphs and Edge-clique graphs

AbstractNot every edge-clique graph is a clique graph. 1 IntroductionLet Gbe a graph. The clique graph of G, K(G), is the intersection graph of thefamily of all maximal cliques of G. The edge-clique graph of G, K e (G), is theone whose vertices are the edges of G, two vertices being adjacent in K e (G),when the corresponding edges of G belong to a same clique. A graph G is aclique graph (edge-clique graph) if there exist a graph H such that K(H) = G(K e (H) = G).In a 1991 paper [4], Theorem 1, it is affirmed that every edge-clique graph isa clique graph (*). However, Prisner [1] noted that the proof of Theorem 1 isnot correct. In this note we shall prove that (*) does not hold, i.e. we showthat it is not always the case where an edge-clique graph is a clique graph.In Section 2 we review some properties of both clique and edge-clique graphsand observe that every edge-clique graph whose largest clique has size at mostthree is a clique graph. In Section 3 we show an edge-clique graph that is nota clique graph and prove that it is a minimum counterexample to (*).All graphs considered are finite, simple and undirected. The vertex and edgesets of a graph G are represented by V(G) and E(G), respectively. For C ⊆V(G), say that C is a clique when C induces a complete subgraph in G. Amaximal clique is one not properly contained in any other. Let ω(G) denote

Tree loop graphs

Many problems involving DNA can be modeled by families of intervals. However, traditional interval graphs do not take into account the repeat structure of a DNA molecule. In the simplest case, one repeat with two copies, the underlying line can be seen as folded into a loop. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and, in the affirmative case, produces a loop representation for the tree.

Linear-Interval Dimension and PI Orders

Abstract A PI graph G is the intersection graph of a family of triangles ABC between two distinct parallel lines L1 and L2, such that A is on L1 and B C ¯ is on L2. We study the orders defined by transitive orientations of the complement of G, the PI orders. We describe a characterization for such orders in terms of a special order dimension called linear-interval dimension. We show that the linear-interval dimension of an order is a comparability invariant, which generalizes the well-known result that the interval dimension is a comparability invariant.

On λ-coloring split, chordal bipartite and weakly chordal graphs

Abstract A λ-coloring, or L(2, 1)-coloring, of a graph is an assignment of nonnegative integers to its vertices such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. Given a graph G, λ ( G ) is the minimum range of colors for which there exists a λ-coloring of G. A conjecture by Griggs and Yeh (SIAM J. Discrete Math. 5 (1992), 586–595) states that λ ( G ) is at most Δ 2 , where Δ is the maximum degree of a vertex in G. We prove that this conjecture holds for weakly chordal graphs. Furthermore, we improve the known upper bounds for λ for chordal bipartite graphs and split graphs.