C.M.H. de Figueiredo

On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs

Golumbic, Kaplan, and Shamir, in their paper [M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, J. Algorithms 19 (1995) 449-473] on graph sandwich problems published in 1995, left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open. We prove that the sandwich problem for strongly chordal graphs is NP-complete. We also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs.

Clique graph recognition is NP-complete

A complete set of a graph G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. Denote by the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of . Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. We prove that the clique graph recognition problem is NP-complete.

Total chromatic number of {square,unichord}-free graphs

Abstract We determine a surprising class for which edge-colouring is NP-complete but whose graphs are all Type 1. The class of unichord-free graphs was recently studied by Trotignon and Vuskovic [N. Trotignon and K. Vuskovic. A structure theorem for graphs with no cycle with a unique chord and its consequences . To appear in J. Graph Theory. (Available online at http://www.comp.leeds.ac.uk/vuskovi/chord.ps .)] in the context of vertex-colouring. Machado, Figueiredo and Vuskovic [R. C. S. Machado, C. M. H. de Figueiredo, and K. Vuskovic. Chromatic index of graphs with no cycle with unique chord . In: Proc. of the ALIO/EUROWorkshop on Applied Optimization (2008). Full paper submitted to Theoret. Comput. Sci.] established the NP-completeness of edge-colouring unichord-free graphs. For the subclass of {square,unichord}-free graphs, an interesting complexity dicothomy holds [R. C. S. Machado, C. M. H. de Figueiredo, and K. Vuskovic. Chromatic index of graphs with no cycle with unique chord . In: Proc. of the ALIO/EUROWorkshop on Applied Optimization (2008). Full paper submitted to Theoret. Comput. Sci.]: if the maximum degree is 3, the edge-colouring is NP-complete, otherwise, the problem is polynomial. Subsequently, Machado and Figueiredo [R. C. S. Machado and C. M. H. de Figueiredo. Total chromatic number of chordless graphs . In: Proc. of the Cologne-Twente Workshop (2009). Full paper submitted to Discrete Appl. Math.] settled the validity of the Total-Colouring Conjecture for {square,unichord}-free graphs by proving that non-complete {square,unichord}-free graphs of maximum degree at least 4 are Type 1. In the present work, we prove that non-complete {square,unichord}-free graphs of maximum degree 3 are Type 1, establishing the polynomiality of total-colouring restricted to {square,unichord}-free graphs.

Skew partition sandwich problem is NP-complete

Abstract Sandwich problems generalize graph recognition problems with respect to a property Π. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for a third graph, required to satisfy a property Π, whose edge set lies between the edge sets of two given graphs. A skew partition of a graph G = ( V , E ) is a partition of its vertex set V into four nonempty parts A, B, C, D such that each vertex of part A is adjacent to each vertex of part B, and each vertex of part C is nonadjacent to each vertex of part D. Skew cutset generalizes star cutset which in turn generalizes both homogeneous set and clique cutset. Homogeneous set, clique cutset, star cutset, and skew cutset are decompositions arising in perfect graph theory and the recognition of each decomposition is known to be polynomial. Regarding sandwich problems, it is known that homogeneous set sandwich problem is polynomial, clique cutset sandwich problem is NP-complete, and star cutset sandwich problem is polynomial. We prove that skew partition sandwich problem is NP-complete, establishing an interesting computational complexity non-monotonicity.

The total chromatic number of split-indifference graphs ☆

Abstract The total chromatic number of a graph G , χ T ( G ) , is the least number of colours sufficient to colour the vertices and edges of a graph such that no incident or adjacent elements (vertices or edges) receive the same colour. The Total Colouring Conjecture (TCC) states that every simple graph G has χ T ( G ) ≤ Δ ( G ) + 2 , and it is a challenging open problem in Graph Theory. For both split graphs and indifference graphs, the TCC holds, and χ T ( G ) = Δ ( G ) + 1 when Δ ( G ) is even. For a split-indifference graph G with odd Δ ( G ) , we give conditions for its total chromatic number to be Δ ( G ) + 2 , and we build a ( Δ ( G ) + 1 ) -total colouring otherwise. Also, we pose a conjecture for a class of graphs that generalizes split-indifference graphs.

On Coloring Problems of Snark Families

Abstract Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Theorem. In 2003, Cavicchioli et al. proved that for snarks with less than 30 vertices, the total chromatic number is 4, and proposed the problem of finding (if any) the smallest snark which is not 4-total colorable. Since then, only two families of snarks have had their total chromatic number determined to be 4, namely the Flower Snark family and the Goldberg family. We prove that the total chromatic number of the Loupekhine family is 4. We study the dot product, a known operation to construct snarks. We consider families of snarks using the dot product, particularly subfamilies of the Blanusa families, and obtain a 4-total coloring for each family. We study edge coloring properties of girth trivial snarks that cannot be extended to total coloring. We classify the snark recognition problem as CoNP-complete and establish that the chromatic number of a snark is 3.

The cost of perfection for matchings in graphs

Abstract Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs.

On the equitable total chromatic number of cubic graphs

A total coloring is equitable if the number of elements colored with each color differs by at most one, and the least integer for which a graph has such a coloring is called its equitable total chromatic number. Wang conjectured that the equitable total chromatic number of a multigraph of maximum degree is at most + 2 , and proved this for the case where 3 . Therefore, the equitable total chromatic number of a cubic graph is either 4 or 5, and in this work we prove that the problem of deciding whether it is 4 is NP-complete for bipartite cubic graphs.Furthermore, we present the first known Type1 cubic graphs with equitable total chromatic number5. All of them have, by construction, a small girth. We also find one infinite family of Type1 cubic graphs of girth5 having equitable total chromatic number 4. This motivates the following question: Does there exist Type1 cubic graphs of girth greater than5 and equitable total chromatic number5?

Using SPQR-trees to speed up recognition algorithms based on 2-cutsets ☆

Abstract Several well-studied classes of graphs admit structural characterizations via proper 2-cutsets which lead to polynomial-time recognition algorithms. The algorithms so far obtained for those recognition problems do not guarantee linear-time complexity. The bottleneck to those algorithms is the Ω ( n m ) -time complexity to fully decompose by proper 2-cutsets a graph with n vertices and m edges. In the present work, we investigate the 3-connected components of a graph and propose the use of the SPQR-tree data structure to obtain a fully decomposed graph in linear time. As a consequence, we show that the recognition of chordless graphs and of graphs that do not contain a propeller as a subgraph can be done in linear time, answering questions in the existing literature.

Using SPQR-trees to speed up algorithms based on 2-cutset decompositions

Abstract We propose the use of SPQR-trees as a data structure to encode the 3-connected components of a graph and to obtain linear-time recognition algorithms for graph classes structurally characterized by 2-cutset decompositions.