Bull-free Graphs Research Articles

Odd Holes in Bull-Free Graphs

The complexity of testing whether a graph contains an induced odd cycle of length at least five is currently unknown. In this paper we show that this test can be done in polynomial time if the inpu...

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time 2O(k5)⋅n9. In this article we improve this running time to 2O(k2)⋅n7. As a byproduct, we also improve the previous Turing-kernel for this problem from O(k5) to O(k2). Furthermore, for the subclass of bull-free graphs without holes of length at most 2p−1 for p≥3, we speed up the running time to 2O(k⋅k1p−1)⋅n7. As p grows, this running time is asymptotically tight in terms of k, since we prove that for each integer p≥3, Weighted Independent Set cannot be solved in time 2o(k)⋅nO(1) in the class of {bull,C4,…,C2p−1}-free graphs unless the ETH fails.

The complexity of partitioning into disjoint cliques and a triangle-free graph

Motivated by Chudnovsky’s structure theorem of bull-free graphs, Abu-Khzam, Feghali, and Müller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs.

A Polynomial Turing-Kernel for Weighted Independent Set in Bull-Free Graphs

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size $k$, when $k$ is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size $k$. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turing-kernel. More precisely, the hard cases are instances of size $O(k^5)$. As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem of bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.

Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull

The maximum weight independent set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as for odd-hole- and bull-free graphs (dart and bull have five vertices, say \(a,b,c,d,e\), and dart has edges \(ab,ac,ad,bd,cd,de\), while bull has edges \(ab,bc,cd,be,ce\)). If the graphs are hole-free instead of odd-hole-free then stronger structural results and better time bounds are obtained.

The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem

The Π graph sandwich problem asks, for a pair of graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2, whether there exists a graph G=(V,E) that satisfies property Π and E1⊆E⊆E2. We consider the property of being F-free, where F is a fixed graph. We show that the claw-free graph sandwich and the bull-free graph sandwich problems are both NP-complete, but the paw-free graph sandwich problem is polynomial. This completes the study of all cases where F has at most four vertices. A skew partition of a graph G is a partition of its vertex set into four nonempty parts A,B,C,D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is nonadjacent to each vertex of D. We prove that the skew partition sandwich problem is NP-complete, establishing a computational complexity non-monotonicity.

Simplicial Vertices in Graphs with no Induced Four-Edge Path or Four-Edge Antipath, and theH6-Conjecture

Let be the class of all graphs with no induced four-edge path or four-edge antipath. Hayward and Nastos [6] conjectured that every prime graph in not isomorphic to the cycle of length five is either a split graph or contains a certain useful arrangement of simplicial and antisimplicial vertices. In this article, we give a counterexample to their conjecture, and prove a slightly weaker version. Additionally, applying a result of the first author and Seymour [1] we give a short proof of Fouquet's result [3] on the structure of the subclass of bull-free graphs contained in .

Some properties of edge intersection graphs of single-bend paths on a grid

In this paper we consider graphs G whose vertices can be represented as single-bend paths (i.e., paths with at most one turn) on a rectangular grid, such that two vertices are adjacent in G if and only if the corresponding paths share at least one edge of the grid. These graphs, called B 1 -EPG graphs, were first introduced in Golumbic et al. (2009) [13]. Here we show that the neighborhood of every vertex in a B 1 -EPG graph induces a weakly chordal graph. From this we conclude that the family F of B 1 -EPG graphs satisfies the Erdős–Hajnal property with ϵ ( F ) = 1 3 , i.e., that every B 1 -EPG graph on n vertices contains either a clique or a stable set of size at least n 1 3 . Finally we give a characterization of B 1 -EPG graphs among some subclasses of chordal graphs, namely chordal bull-free graphs, chordal claw-free graphs, chordal diamond-free graphs, and special cases of split graphs.

The structure of bull-free graphs I—Three-edge-paths with centers and anticenters

The bull is the graph consisting of a triangle and two disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is the first paper in a series of three. The goal of the series is to explicitly describe the structure of all bull-free graphs. In this paper we study the structure of bull-free graphs that contain as induced subgraphs three-edge-paths P and Q, and vertices c ∉ V ( P ) and a ∉ V ( Q ) , such that c is adjacent to every vertex of V ( P ) and a has no neighbor in V ( Q ) . One of the theorems in this paper, namely 1.2, is used in Chudnovsky and Safra (2008) [9] in order to prove that every bull-free graph on n vertices contains either a clique or a stable set of size n 1 4 , thus settling the Erdös–Hajnal conjecture (Erdös and Hajnal, 1989) [17] for the bull.

The structure of bull-free graphs II and III—A summary

The bull is a graph consisting of a triangle and two pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is a summary of the last two papers [2,3] in a series [1–3] (Chudnovsky, 2012). The goal of the series is to give a complete description of all bull-free graphs. We call a bull-free graph elementary if it does not contain an induced three-edge-path P such that some vertex c ∉ V ( P ) is complete to V ( P ) , and some vertex a ∉ V ( P ) is anticomplete to V ( P ) . Here we prove that every elementary graph either belongs to one of a few basic classes, or admits a certain decomposition, and then uses this result together with the results of [1] ( this issue) to give an explicit description of the structure of all bull-free graphs.

The Erdős–Hajnal conjecture for bull-free graphs

The bull is a graph consisting of a triangle and two pendant edges. A graphs is called bull-free if no induced subgraph of it is a bull. In this paper we prove that every bull-free graph on n vertices contains either a clique or a stable set of size n 1 4 , thus settling the Erdős–Hajnal conjecture [P. Erdős, A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989) 37–52] for the bull.

On the Structure of Bull-Free Perfect Graphs, 2: the Weakly Chordal Case

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. A graph is perfectly orderable if it admits an ordering such that the greedy sequential method applied on this ordering produces an optimal coloring for every induced subgraph. Chvatal conjectured that every bull-free graph with no odd hole or antihole is perfectly orderable. In a previous paper we studied the structure of general bull-free perfect graphs, and reduced Chvatal's conjecture to the case of weakly chordal graphs. Here we focus on weakly chordal graphs, and we reduce Chvatal's conjecture to a restricted case. Our method lays out the structure of all bull-free weakly chordal graphs. These results have been used recently by Hayward to establish Chvatal's conjecture for this restricted case and therefore in full.

Bull-Free Weakly Chordal Perfectly Orderable Graphs

Using doubly lexical orders and the notion of box partition due to de Figueiredo, Maffray, and Porto, we show that a certain subclass of bull-free weakly chordal graphs is perfectly orderable. This together with results of de Figueiredo, Maffray, and Porto confirms Chvatal's conjecture that bull-free graphs with no anti-hole and no odd hole are perfectly orderable; here hole means induced cycle with five or more vertices.

On the structure of bull-free perfect graphs

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvatal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphs

Recognizing bull-free perfect graphs

A bull is the graph obtained from a triangle by adding two pendant vertices adjacent to distinct vertices of the triangle. Chvatal and Sbihi showed that the strong perfect graph conjecture holds for Bull-free graphs. We give a polynomial time recognition algorithm for Bull-free perfect graphs.