Luerbio Faria

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.

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.

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.

Splitting Number is NP-complete

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k ≥ 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v either to v1 or to v2. We prove that the splitting number decision problem is NP-complete, even when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs also implies NP-completeness for graphs not containing a subdivision of K5 as a subgraph.

The complexity of clique graph recognition

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C(G) the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of C(G). 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. However, the time complexity of the problem of recognizing clique graphs is a long-standing open question. We prove that the clique graph recognition problem is NP-complete.

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.

Odd cycle transversals and independent sets in fullerene graphs

A fullerene graph is a cubic bridgeless plane graph with all faces of size

Nonplanar vertex deletion: maximum degree thresholds for NP/Max SNP-hardness and a 34-approximation for finding maximum planar induced subgraphs☆

5

Unitary Toric Classes, the Reality and Desire Diagram, and Sorting by Transpositions

and

Oriented coloring in planar, bipartite, bounded degree 3 acyclic oriented graphs

6