Van Bang Le

Graph classes: a survey

Preface 1. Basic Concepts 2. Perfection, Generalized Perfection, and Related Concepts 3. Cycles, Chords and Bridges 4. Models and Interactions 5. Vertex and Edge Orderings 6. Posets 7. Forbidden Subgraphs 8. Hypergraphs and Graphs 9. Matrices and Polyhedra 10. Distance Properties 11. Algebraic Compositions and Recursive Definitions 12. Decompositions and Cutsets 13. Threshold Graphs and Related Concepts 14. The Strong Perfect Graph Conjecture Appendix A. Recognition Appendix B. Containment Relationships Bibliography Index.

The complexity of some problems related to Graph 3-COLORABILITY

Abstract It is well-known that the Graph 3-colorability problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in lt]o li](1) a tree or li](2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given graph has a stable set whose deletion results in li](3) the complement of a bipartite graph, li](4) a split graph or li](5) a threshold graph.

Structure and linear time recognition of 3-leaf powers

A graph G is the k-leaf power of a tree T if its vertices are leaves of T such that two vertices are adjacent in G if and only if their distance in T is at most k. Then T is the k-leaf root of G. This notion was introduced and studied by Nishimura, Ragde, and Thilikos motivated by the search for underlying phylogenetic trees. Their results imply a O(n^3) time recognition algorithm for 3-leaf powers. Later, Dom, Guo, Huffner, and Niedermeier characterized 3-leaf powers as the (bull,@?dart,@?gem)-free chordal graphs. We show that a connected graph is a 3-leaf power if and only if it results from substituting cliques into the vertices of a tree. This characterization is much simpler than the previous characterizations via critical cliques and forbidden induced subgraphs and also leads to linear time recognition of these graphs.

Structure and linear-time recognition of 4-leaf powers

A graph <i>G</i> is the <i>k-leaf power</i> of a tree <i>T</i> if its vertices are leaves of <i>T</i> such that two vertices are adjacent in <i>G</i> if and only if their distance in <i>T</i> is at most <i>k</i>. Then <i>T</i> is a <i>k-leaf root</i> of <i>G</i>. This notion was introduced and studied by Nishimura, Ragde, and Thilikos [2002], motivated by the search for underlying phylogenetic trees. Their results imply an <i>O</i>(<i>n</i>³)-time recognition algorithm for 4-leaf powers. Recently, Rautenbach [2006] as well as Dom et al. [2005] characterized 4-leaf powers without true twins in terms of forbidden subgraphs. We give new characterizations for 4-leaf powers and squares of trees by a complete structural analysis. As a consequence, we obtain a conceptually simple linear-time recognition of 4-leaf powers.

Tree spanners on chordal graphs: complexity and algorithms

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The TREE t-SPANNER problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359-387) by showing that, for any t ≥ 4, TREE t-SPANNER is NP-complete even on chordal graphs of diameter at most t + 1 (if t is even), respectively, at most t + 2 (if t is odd). Then we point out that every chordal graph of diameter at most t - 1 (respectively, t - 2) admits a tree t-spanner whenever t ≥ 2 is even (respectively, t ≥ 3 is odd), and such a tree spanner can be constructed in linear time.The complexity status of TREE 3-SPANNER still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide TREE 3-SPANNER efficiently.

On stable cutsets in graphs

We answer a question of Corneil and Fonlupt by showing that deciding whether a graph has a stable cutset is NP-complete even for restricted graph classes. Some efficiently solvable cases will be discussed, too.

On the complexity of 4-coloring graphs without long induced paths

We show that deciding if a graph without induced paths on nine vertices can be colored with 4 colors is an NP-complete problem, improving a previous NP-completeness result proved by Woeginger and Sgall in 2001. The complexity of 4-coloring graphs without induced paths on five vertices remains open. We show that deciding if a graph without induced paths or cycles on five vertices can be colored with 4 colors can be done in polynomial time. A step in our algorithm uses the well-known and deep fact due to Grotschel, Lovasz and Schrijver stating that perfect graphs can be optimally colored in polynomial time.

Stability number of bull- and chair-free graphs revisited

De Simone showed that prime bull- and chair-free graphs containing a co-diamond are either bipartite or an induced cycle of odd length at least five. Based on this result, we give a complete structural characterization of prime (bull,chair)-free graphs having stability number at least four as well as of (bull,chair,co-chair)-free graphs. This implies constant-bounded clique width for these graph classes which leads to linear time algorithms for some algorithmic problems. Moreover, we obtain a robust O(nm) time algorithm for the maximum weight stable set problem on bull-and chair-free graphs without testing whether the (arbitrary) input graph is bull- and chair-free. This improves previous results with respect to structural insight, robustness and time bounds.

Optimal tree 3-spanners in directed path graphs

In a graph G, a spanning tree T is called a tree t-spanner of G if the distance between any two vertices in T is at most t times their distance in G. While the complexity of finding a tree t-spanner of a given graph is known for any fixed t fi 3, the case t 5 3 still remains open. In this article, we show that each directed path graph G has a tree 3-spanner T by means of a linear-time algorithm constructing T. Moreover, the output tree 3-spanner T is optimal in the sense that G has a tree 2-spanner if and only if T is a tree 2-spanner of G.

On stable cutsets in line graphs

We answer a question of Brandstadt et al. by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is NP-complete. Conversely, the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently. The proof of our NP-completeness result is based on a refinement on a result due to Chvatal that recognizing decomposable graphs with maximum degree 4 is an NP-complete problem. Here, a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v. We also discuss some open problems on stable cutsets.