Célia Picinin de Mello

A linear-time algorithm for proper interval graph recognition

Interval graphs are the intersection graphs of families of intervals in the real line. If the intervals can be chosen so that no interval contains another, we obtain the subclass of proper interval graphs. We show how to recognize proper interval graphs in linear time by constructing the clique partition from the output of a single lexicographic breadth-first search.

On clique-complete graphs

Abstract The clique graph, K(G) , of a graph G is the intersection graph of the maximal cliques of G . For a natural number n , a graph G is n-convergent if K n ( G ) is isomorphic to K 1 (the one-vertex graph). A graph G is convergent if it is n-convergent for some natural number n . A 2-convergent graph is called clique - complete . We describe the family of minimal graphs which are clique-complete but have no universal vertices. The minimality used here refers to induced subgraphs. In addition, we show that recognizing clique-complete graphs is Co-NP-complete.

Total-chromatic number and chromatic index of dually chordal graphs

Abstract Given a graph G and a vertex v , a vertex u∈N(v) is a maximum neighbor of v if for all w∈N(v) we have N(w)⫅N(u) , where N(v) denotes the neighborhood of v in G . A maximum neighborhood elimination order of G is a linear order v 1 ,v 2 ,…,v n on its vertex set such that there is a maximum neighbor of v i in the subgraph G[v 1 ,…,v i ] . A graph is dually chordal if it admits a maximum neighborhood elimination order. Alternatively, a graph is dually chordal if it is the clique graph of a chordal graph. The class of dually chordal graphs generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs, interval graphs, and indifference graphs. We prove that Vizings total-color conjecture holds for dually chordal graphs. We describe a new heuristic that yields an exact total coloring for even maximum degree dually chordal graphs and an exact edge coloring for odd maximum degree dually chordal graphs.

Even and odd pairs in comparability and in P 4 -comparability graphs

We characterize even and odd pairs in comparability and in P4-comparability graphs. The characterizations lead to simple algorithms for deciding whether a given pair of vertices forms an even or odd pair in these classes of graphs. The complexities of the proposed algorithms are O(n + m) for comparability graphs and O(n2m) for P4-comparability graphs. The former represents an improvement over a recent algorithm of complexity O(nm).

Sources and Sinks in Comparability Graphs

We prove that a subset S of vertices of a comparability graph G is a source set if and only if each vertex of S is a source and there is no odd induced path in G between two vertices of S. We also characterize pairs of subsets corresponding to sources and sinks, respectively. Finally, an application to interval graphs is obtained.

Decompositions for the edge colouring of reduced indifference graphs

The chromatic index problem--finding the minimum number of colours required for colouring the edges of a graph--is still unsolved for indifference graphs, whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order. We present new positive evidence for the conjecture: every non neighbourhood-overfull indifference graph can be edge coloured with maximum degree colours. Two adjacent vertices are twins if they belong to the same maximal cliques. A graph is reduced if it contains no pair of twin vertices. A graph is overfull if the total number of edges is greater than the product of the maximum degree by ⌈n/2⌉, where n is the number of vertices. We give a structural characterization for neighbourhood-overfull indifference graphs proving that a reduced indifference graph cannot be neighbourhood-overfull. We show that the chromatic index for all reduced indifference graphs is the maximum degree. We present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours.

Colouring Clique-Hypergraphs of Circulant Graphs

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph,

Recognizing Well Covered Graphs of Families with Special P4-Components

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