Recognizing HH-free, HHD-free, and Welsh-Powell Opposition Graphs

2011

An (h, s, t)-representation of a graph G consists of a collection of subtrees of a tree T , where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at mots s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that have an (h, s, t)-representation is denoted [h, s, t]. An undirected graph G is called a V P T graph if it is the vertex intersection graph of a family of paths in a tree. In this paper we characterize [h, 2, 1] graphs using chromatic number. We show that the problem of deciding whether a given V P T graph belongs to [h, 2, 1] is NP-complete, while the problem of deciding whether the graph belongs to [h, 2, 1] − [h − 1, 2, 1] is NP-hard. Both problems remain hard even when restricted to Split ∩ V P T. Additionally, we present a non-trivial subclass of Split ∩ V P T in which these problems are polynomial time solvable.

Journal of Graph Theory, 2000

We present an algorithm that determines in polytime whether a graph contains an even hole. The algorithm is based on a decomposition theorem for even-hole-free graphs obtained in Part I of this paper. We a l s o g i v e a polytime algorithm to nd an even hole in a graph when one exists.

European Journal of Combinatorics, 1996

This paper contains a new algorithm that recognizes whether a given graph G is a Hamming graph , i . e . a Cartesian product of complete graphs , in O ( m ) time and O ( n 2 ) space . Here m and n denote the numbers of edges and vertices of G , respectively . Previously this was only possible in O ( m log n ) time .

Discrete Applied Mathematics, 2003

In a graph G =(V; E), the eccentricity e(v) of a vertex v is max{d(v; u): u ∈ V }. The center of a graph is the set of vertices with minimum eccentricity. A house-hole-domino-free (HHD-free) graph is a graph which does not contain the house, the domino, and holes (cycles of length at least ÿve) as induced subgraphs. We present an algorithm which ÿnds a central vertex of a HHD-free graph in O( 1:376 |V |) time, where is the maximum degree of a vertex of G. Its complexity is linear in the case of weak bipolarizable graphs, chordal graphs, and distance-hereditary graphs. The algorithm uses special metric and convexity properties of HHD-free graphs. ?

Discrete Applied Mathematics, 2015

A graph is hole-and diamond-free (HD-free) if none of its induced subgraphs is isomorphic to a chordless cycle of length at least five or to a diamond. Using the clique separator approach and the simple structure of atoms of HD-free graphs, we show how to recognize HD-free graphs in time O(n 2). One of the main tools is Lexicographic Breadth-First Search (LexBFS); we give two new properties of LexBFS which are essential for reaching the time bound and which hold for any graph. Moreover, we find minimal triangulations of HD-free graphs in time O(n 2), introducing efficient algorithms for the minimal triangulation of matched co-bipartite graphs and chordal bipartite graphs.

Theoretical Computer Science, 2005

A graph is (P 5 ,gem)-free, when it does not contain P 5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P 4 ) as an induced subgraph.

Journal of Graph Theory, 2018

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove that a (banner, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or contains a homogeneous set. Using this structure result, we design a polynomial-time algorithm for recognizing (banner, odd hole)-free graphs. We also design polynomialtime algorithms to find, for such a graph, a minimum coloring and largest stable set. A graph G is perfectly divisible if every induced subgraph H of G contains a set X of vertices such that X meets all largest cliques of H, and X induces a perfect graph. The chromatic number of a perfectly divisible graph G is bounded by ω 2 where ω denotes the number of vertices in a largest clique of G. We prove that (banner, odd hole)-free graphs are perfectly divisible.

Parallel Processing Letters, 2004

We prove algorithmic characterizations of weakly chordal graphs, which lead to efficient parallel algorithms for recognizing P5-free and [Formula: see text]-free weakly chordal graphs. For an input graph on n vertices and m edges, our algorithms run in O( log 2n) time and require O(m2/ log n) processors on the EREW PRAM model of computation. The proposed recognition algorithms efficiently detect P5 s and [Formula: see text] in weakly chordal graphs in O( log n) time with O(m2/ log n) processors on the EREW PRAM. Additionally, we show how the algorithms can be augmented to provide a certificate for the existence of a P5 (or a [Formula: see text]) in case the input graph is not P5-free (respectively, [Formula: see text]-free) weakly chordal.

2003

A graph is (P 5 ,gem)-free, when it does not contain P 5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P 4 ) as an induced subgraph.

Journal of Graph Theory, 2017

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our O(nm)-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our O(n 2.5 + nm)-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.

Graph-Theoretic Concepts in Computer Science, 2021

For a class G of graphs, the problem Subgraph Complement to G asks whether one can find a subset S of vertices of the input graph G such that complementing the subgraph induced by S in G results in a graph in G. We investigate the complexity of the problem when G is H-free for H being a complete graph, a star, a path, or a cycle. We obtain the following results: • When H is a K t (a complete graph on t vertices) for any fixed t ≥ 1, the problem is solvable in polynomial-time. This applies even when G is a subclass of K t-free graphs recognizable in polynomial-time, for example, the class of (t − 2)-degenerate graphs. • When H is a K 1,t (a star graph on t + 1 vertices), we obtain that the problem is NP-complete for every t ≥ 5. This, along with known results, leaves only two unresolved cases-K 1,3 and K 1,4. • When H is a P t (a path on t vertices), we obtain that the problem is NP-complete for every t ≥ 7, leaving behind only two unresolved cases-P 5 and P 6. • When H is a C t (a cycle on t vertices), we obtain that the problem is NP-complete for every t ≥ 8, leaving behind four unresolved cases-C 4 , C 5 , C 6 , and C 7. Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time 2 o(|V (G)|)), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for G are applicable for G, thereby obtaining similar results for H being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).

2014

In the chapter on decomposition trees we start with an explanation of the graph minor theory. As a basic example we show that this implies that feedback vertex set is fixed-parameter tractable. Next, we introduce treewidth as a parametrization of chordal graphs. We show that the class of graphs of bounded treewidth is closed under minors and, as a consequence, that the class can be recognized in O(n 2 ) time. We give an easy linear-time algorithm for the recognition of graphs with treewidth two. For general k we explain the historic O(n k+2 ) algorithm of Arnborg, Corneil and Proskurowski, for the recognition of partial k-trees. We introduce tree decompositions as the clique trees of chordal embeddings of a graph. Likewise, we introduce rank decompositions as a parametrization of the decomposition trees for distancehereditary graphs. We close the chapter with a brief discussion of monadic second-order logic.

Fundamenta Informaticae, 2016

We consider a new graph operation c 2-join which generalizes join and co-join. We show that odd hole-free graphs (odd antihole-free graphs) are closed under c 2-join and describe a polynomial time algorithm to recognize graphs that admit a c 2-join. The time complexity of the (a) recognition problem, (b) maximum weight independent set (MWIS) problem, and (c) minimum coloring (MC) problem for odd hole-free graphs are still unknown. Let H be an odd hole-free graph that contains an odd antihole as an induced subgraph and G H be the class of all graphs generated from the induced subgraphs of H by using c 2-join recursively. Then G H is odd hole-free, contains all P 4-free graphs, complement of all bipartite graphs, and some imperfect graphs. We show that the MWIS problem, maximum weight clique (MWC) problem, MC problem, and minimum clique cover (MCC) problem can be solved efficiently for G H .

2006

2016

We establish the complexity of several graph embedding problems: Subgraph Isomorphism, Graph Minor, Induced Subgraph and Induced Minor, when restricted to H-minor free graphs. In each of these problems, we are given a pattern graph P and a host graph G, and want to determine whether P is a subgraph (minor, induced subgraph or induced minor) of G. We show that, for any fixed graph H and epsilon > 0, if P is H-Minor Free and G has treewidth tw, (induced) subgraph can be solved 2^{O(k^{epsilon}*tw+k/log(k))}*n^{O(1)} time and (induced) minor can be solved in 2^{O(k^{epsilon}*tw+tw*log(tw)+k/log(k))}*n^{O(1)} time, where k = |V(P)|. We also show that this is optimal, in the sense that the existence of an algorithm for one of these problems running in 2^{o(n/log(n))} time would contradict the Exponential Time Hypothesis. This solves an open problem on the complexity of Subgraph Isomorphism for planar graphs. The key algorithmic insight is that dynamic programming approaches can be spe...