Simone Dantas

On decision and optimization ( k, l )-graph sandwich problems

A graph G is (k,l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k,l)-Graph Sandwich Problem asks, given two graphs G1 = (V,E1) and G2 = (V,E2), whether there exists a graph G = (V,E) such that E1 ⊆ E ⊆ E2 and G is (k,l). In this paper, we prove that the (k,l)-Graph Sandwich Problem is NP-complete for the cases k=1 and l=2; k=2 and l=1; or k=l=2. This completely classifies the complexity of the (k,l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+l > 2; the problem is polynomial otherwise. We consider the degree Δ constraint subproblem and completely classify the problem as follows: the problem is polynomial, for k ≤ 2 or Δ ≤ 3; the problem is NP-complete otherwise. In addition, we propose two optimization versions of graph sandwich problem for a property Π: MAX-Π-GSP and MIN-Π-GSP. We prove that MIN-(2,1)-GSP is a Max-SNP-hard problem, i.e., there is a positive constant e, such that the existence of an e-approximative algorithm for MIN-(2,1)-GSP implies P = NP.

FINDING H-PARTITIONS EFFICIENTLY ∗

We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, 4-part, external constraints only (no internal constraints), each part non-empty. We describe tools that yield for each problem considered in this paper a simple and low complexity polynomial-time algorithm. Mathematics Subject Classification. 05C85, 68R10.

More fires and more fighters

Hartnells firefighter game models the containment of the spreading of an undesired property within a network. It is a one-player game played in rounds on a graph G in which a fire breaks out at f vertices of G. In each round the fire spreads to neighboring vertices unless the player defends these. The power of the player is limited in the sense that he can defend at most d additional vertices of G in each round. His objective is to save as many vertices as possible from burning. Most research on this game concerned the case f=d=1, which already leads to hard problems even restricted to trees. We study the game for larger values of f and d. We present useful properties of optimal strategies for the game on trees, efficient approximation algorithms, and bounds on the so-called surviving rate.

The polynomial dichotomy for three nonempty part sandwich problems

We classify into polynomial time or NP-complete all three nonempty part sandwich problems. This solves the polynomial dichotomy into polynomial time and NP-complete for this class of graph partition problems.

Colouring clique-hypergraphs of circulant graphs

Abstract 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, H ( G ) , of a graph G has V ( G ) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of H ( G ) is a clique-colouring of G. Determining the clique-chromatic number, the least number for which a graph G admits a clique-colouring, is known to be NP-hard. We establish that the clique-chromatic number for powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two.

Stable skew partition problem

A skew partition is a partition of the vertex set of a graph into four nonempty parts A,B, C,D such that there are all possible edges between A and B, and no edges between C and D. A stable skew partition is a skew partition where A induces a stable set of the graph. We show that determining if a graph permits a stable skew partition is NP-complete. We discuss limits of such reductions by adding cardinality constraints.

DCJ-indel and DCJ-substitution distances with distinct operation costs

BackgroundClassical approaches to compute the genomic distance are usually limited to genomes with the same content and take into consideration only rearrangements that change the organization of the genome (i.e. positions and orientation of pieces of DNA, number and type of chromosomes, etc.), such as inversions, translocations, fusions and fissions. These operations are generically represented by the double-cut and join (DCJ) operation. The distance between two genomes, in terms of number of DCJ operations, can be computed in linear time. In order to handle genomes with distinct contents, also insertions and deletions of fragments of DNA – named indels – must be allowed. More powerful than an indel is a substitution of a fragment of DNA by another fragment of DNA. Indels and substitutions are called content-modifying operations. It has been shown that both the DCJ-indel and the DCJ-substitution distances can also be computed in linear time, assuming that the same cost is assigned to any DCJ or content-modifying operation.ResultsIn the present study we extend the DCJ-indel and the DCJ-substitution models, considering that the content-modifying cost is distinct from and upper bounded by the DCJ cost, and show that the distance in both models can still be computed in linear time. Although the triangular inequality can be disrupted in both models, we also show how to efficiently fix this problem a posteriori.

DCJ-indel distance with distinct operation costs

The double-cut and join (DCJ) is a genomic operation that generalizes the typical mutations to which genomes are subject. The distance between two genomes, in terms of number of DCJ operations, can be computed in linear time. More powerful is the DCJ-indel model, which handles genomes with unequal contents, allowing, besides the DCJ operations, the insertion and/or deletion of pieces of DNA --- named indel operations. It has been shown that the DCJ-indel distance can also be computed in linear time, assuming that the same cost is assigned to any DCJ or indel operation. In the present work we consider a new DCJ-indel distance in which the indel cost is distinct from and upper bounded by the DCJ cost. Considering that the DCJ cost is equal to 1, we set the indel cost equal to a positive constant w≤1 and show that the distance can still be computed in linear time. This new distance generalizes the previous DCJ-indel distance considered in the literature (which uses the same cost for both types of operations).

Extremal graphs for the list-coloring version of a theorem of Nordhaus and Gaddum

We characterize the graphs G such that Ch(G)+Ch(G-)=n+1, where Ch(G) is the choice number (list-chromatic number) of G and n is its number of vertices.

Restricted DCJ-indel model: sorting linear genomes with DCJ and indels

BackgroundThe double-cut-and-join (DCJ) is a model that is able to efficiently sort a genome into another, generalizing the typical mutations (inversions, fusions, fissions, translocations) to which genomes are subject, but allowing the existence of circular chromosomes at the intermediate steps. In the general model many circular chromosomes can coexist in some intermediate step. However, when the compared genomes are linear, it is more plausible to use the so-called restricted DCJ model, in which we proceed the reincorporation of a circular chromosome immediately after its creation. These two consecutive DCJ operations, which create and reincorporate a circular chromosome, mimic a transposition or a block-interchange. When the compared genomes have the same content, it is known that the genomic distance for the restricted DCJ model is the same as the distance for the general model. If the genomes have unequal contents, in addition to DCJ it is necessary to consider indels, which are insertions and deletions of DNA segments. Linear time algorithms were proposed to compute the distance and to find a sorting scenario in a general, unrestricted DCJ-indel model that considers DCJ and indels.ResultsIn the present work we consider the restricted DCJ-indel model for sorting linear genomes with unequal contents. We allow DCJ operations and indels with the following constraint: if a circular chromosome is created by a DCJ, it has to be reincorporated in the next step (no other DCJ or indel can be applied between the creation and the reincorporation of a circular chromosome). We then develop a sorting algorithm and give a tight upper bound for the restricted DCJ-indel distance.ConclusionsWe have given a tight upper bound for the restricted DCJ-indel distance. The question whether this bound can be reduced so that both the general and the restricted DCJ-indel distances are equal remains open.