Weak Bipolarizable Graphs

We characterize a new class of perfectly orderable graphs and give a polynomial-time recognition algorithm, together with linear-time optimization algorithms for this class of graphs

A tree representation for P4-sparse graphs

AbstractA graph G is P4-sparse if no set of five vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize both the class of cographs and the class of P4-reducible graphs. We give several characterizations for P4-sparse graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P4-sparse graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time

Some Aspects of the Semi-Perfect Elimination

Several efficient algorithms have been proposed to construct a perfect elimination ordering of the vertices of a chordal graph. We study the behaviour of two of these algorithms in relation to a new concept, namely the semi-perfect elimination ordering, which provides a natural generalization of chordal graphs

Quasi-Brittle Graphs, a New Class of Perfectly Orderable Graphs

A graph G is quasi-brittle if every induced subgraph H of G contains a vertex which is incident to no edge extending symmetrically to a chordless path with three edges in either Hor its complement H¯. The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs. We propose to show that the quasi-brittle graphs are perfectly orderable in the sense of Chvátal: there exists a linear order \u3c on their set of vertices such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a \u3c b and d \u3c c

A note on path domination

We study domination between different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks). We succeeded in characterizing those graphs in which every uv-walk of one particular kind dominates every uv-walk of other specific kind. We thereby obtained new characterizations of standard graph classes like chordal, interval and superfragile graphs.Facultad de Ciencias Exacta

Recognition of some perfectly orderable graph classes

AbstractThis paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P4-simplicial graphs are recognized in O(n3.376) time, improving the previous bounds of O(n4) and O(n5), respectively. Brittle and semi-simplicial graphs are recognized in O(n3) time using a randomized algorithm, and O(n3log2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O(m2). Welsh–Powell opposition graphs are recognized in O(n3) time, improving the previous bound of O(n4). HHP-free graphs and maxibrittle graphs are recognized in O(mn) and O(n3.376) time, respectively

A note on path domination

We study domination between different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks). We succeeded in characterizing those graphs in which every uv-walk of one particular kind dominates every uv-walk of other specific kind. We thereby obtained new characterizations of standard graph classes like chordal, interval and superfragile graphs.Facultad de Ciencias Exacta

Results on perfect graphs

The chromatic number of a graph G is the least number of colours that can be assigned to the vertices of G such that two adjacent vertices are assigned different colours. The clique number of a graph G is the size of the largest clique that is an induced subgraph of G. The notion of perfect graphs was first introduced by Claude Berge in 1960. He defined a graph G to be perfect if the chromatic number of H is equal to the clique number of H for every induced subgraph H C G. He also conjectured that perfect graphs are exactly the class of graphs with no induced odd hole (a chordless odd cycle of greater than or equal to five vertices) or no induced complement of an odd hole, an odd anti-hole. This conjecture, that still remains an open problem, is better known as the Strong Perfect Graph Conjecture (or SPGC). An equivalent statement to SPGC is that minimal imperfect graphs are odd holes and odd anti-holes. Fonlupt conjectured that all minimal imperfect graphs with a minimal cutset that is the union of more than one disjoint clique, must be an odd hole. In this thesis we prove that any hole-free graph G with a minimal cutset C that is the union of vertexdisjoint cliques must have a clique in each component o f G — C that sees all of C. We further prove that minimal imperfect graphs with a minimal cutset that is the union of two disjoint cliques have a hole. Since the introduction of perfectly orderable graphs by Chvdtal in 1984, many classes of perfectly orderable graphs and their recognition algorithms have been identified. Perfectly ordered graphs are those graphs G such that for each induced ordered subgraph H of G, the greedy (or, sequential) colouring algorithm produces an optimal colouring of H. Hohng and Reed previously studied six natural subclasses of perfecdy orderable graphs that are defined by the orientations of the P4 ’s. Four of the six classes can be recognized in polynomial time. The recognition problem for the fifth class has been proven to be NP-complete. In this thesis, we discuss the problem o f recognition for the sixth class, known as one-in-one-out graphs. Also, we consider pyramid-free graphs with the same orientation as one-in-one-out graphs and prove that this class of graphs cannot contain a directed 3-cycle of more than one equivalence class