Time Recognition Algorithm Research Articles

Probe threshold and probe trivially perfect graphs

An undirected graph G = ( V , E ) is a probe C graph if its vertex set can be partitioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E ′ ⊆ N × N such that G ′ = ( V , E ∪ E ′ ) is a C graph. In this article we investigate probe threshold and probe trivially perfect graphs and characterise them in terms of certain 2- Sat formulas and in other ways. For the case when the partition into probes and nonprobes is given, we give characterisations by forbidden induced subgraphs, linear recognition algorithms (in the case of probe threshold graphs it is based on the degree sequence of the graph), and linear algorithms to find a set E ′ of minimum size. Furthermore, we give linear time recognition algorithms for both classes and a characterisation by forbidden subgraphs for probe threshold graphs when the partition ( P , N ) is not given.

The clique operator on circular-arc graphs

A circular-arc graph G is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G . Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graph K ( G ) of a graph G is the intersection graph of its cliques. The iterated clique graph K i ( G ) of G is defined by K 0 ( G ) = G and K i + 1 ( G ) = K ( K i ( G ) ) . In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K -converges, if it is K -convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.

Treelike comparability graphs

An undirected graph is a treelike comparability graph if it admits a transitive orientation such that its transitive reduction is a tree. We show that treelike comparability graphs are distance hereditary. Utilizing this property, we give a linear time recognition algorithm. We then characterize permutation graphs that are treelike. Finally, we consider the Partitioning into Bounded Cliques problem on special subgraphs of treelike permutation graphs.

An Optimal Parallel Algorithm for Constructing a Spanning Tree on Circular Permutation Graphs

The spanning tree problem is to find a tree that connects all the vertices of G. This problem has many applications, such as electric power systems, computer network design and circuit analysis. Klein and Stein demonstrated that a spanning tree can be found in O(log n) time with O(n + m) processors on the CRCW PRAM. In general, it is known that more efficient parallel algorithms can be developed by restricting classes of graphs. Circular permutation graphs properly contain the set of permutation graphs as a subclass and are first introduced by Rotem and Urrutia. They provided O(n2.376) time recognition algorithm. Circular permutation graphs and their models find several applications in VLSI layout. In this paper, we propose an optimal parallel algorithm for constructing a spanning tree on circular permutation graphs. It runs in O(log n) time with O(n/ log n) processors on the EREW PRAM.

Structure and linear-time recognition of 4-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 a k-leaf root of G . This notion was introduced and studied by Nishimura, Ragde, and Thilikos [2002], motivated by the search for underlying phylogenetic trees. Their results imply an O ( n 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.

A characterization of cycle-free unit probe interval graphs

A graph is a probe interval graph (PIG) if its vertices can be partitioned into probes and nonprobes with an interval assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of them is a probe. PIGs are a generalization of interval graphs introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. PIGs have been characterized in the cycle-free case by Sheng, and other miscellaneous results are given by McMorris, Wang, and Zhang. Johnson and Spinrad give a polynomial time recognition algorithm for when the partition of vertices into probes and nonprobes is given. The complexity for the general recognition problem is not known. Here, we restrict attention to the case where all intervals have the same length, that is, we study the unit probe interval graphs and characterize the cycle-free graphs that are unit probe interval graphs via a list of forbidden induced subgraphs.

Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction

A spanning tree T of a graph G is called a tree t - spanner, if the distance between any two vertices in T is at most t -times their distance in G . A graph that has a tree t -spanner is called a tree t - spanner admissible graph. The problem of deciding whether a graph is tree t -spanner admissible is NP-complete for any fixed t ≥ 4 , and is linearly solvable for t = 1 and t = 2 . The case t = 3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a − b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81–87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203–210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.

Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [C.H.C. Little, A characterization of convertible ( 0 , 1 ) -matrices, J. Combin. Theory Ser. B 18 (1975) 187–208] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K 3 , 3 . We relax the notion of a matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion. We define Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize non-Pfaffian factor-critical graphs in terms of forbidden central subgraphs.

On some algorithmic problems regarding the hairpin completion

We consider a few algorithmic problems regarding the hairpin completion. The first problem we consider is the membership problem of the hairpin and iterated hairpin completion of a language. We propose an O ( n f ( n ) ) and O ( n 2 f ( n ) ) time recognition algorithm for the hairpin completion and iterated hairpin completion, respectively, of a language recognizable in O ( f ( n ) ) time. We show that the n factor is not needed in the case of hairpin completion of regular and context-free languages. The n 2 factor is not needed in the case of iterated hairpin completion of context-free languages, but it is reduced to n in the case of iterated hairpin completion of regular languages. We then define the hairpin completion distance between two words and present a cubic time algorithm for computing this distance. A linear time algorithm for deciding whether or not the hairpin completion distance with respect to a given word is connected is also discussed. Finally, we give a short list of open problems which appear attractive to us.

On cliques of Helly Circular-arc Graphs

Abstract A circular-arc graph is the intersection graph of a set of arcs on the circle. It is a Helly circular-arc graph if it has a Helly model, where every maximal clique is the set of arcs that traverse some clique point on the circle. A clique model is a Helly model that identifies one clique point for each maximal clique. A Helly circular-arc graph is proper if it has a Helly model where no arc is a subset of another. In this paper, we show that the clique intersection graphs of Helly circular-arc graphs are related to the proper Helly circular-arc graphs. This yields the first polynomial (linear) time recognition algorithm for the clique graphs of Helly circular-arc graphs. Next, we develop an O ( n ) time algorithm to obtain a clique model of Helly model, improving the previous O ( n 2 ) bound. This gives a linear-time algorithm to find a proper Helly model for the clique graph of a Helly circular-arc graph. As an application, we find a maximum weighted clique of a Helly circular-arc graph in linear time.

Partial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G . A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G . The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem. Recently we were able to characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs. These characterizations lead to polynomial time recognition of clique-perfect graphs in these classes of graphs. In this paper we solve the characterization problem in two new classes of graphs: diamond-free and Helly circular-arc ( HCA ) graphs. This last characterization leads to a polynomial time recognition algorithm for clique-perfect HCA graphs.

Biclique-Helly Graphs

A graph is biclique-Helly when its family of (maximal) bicliques is a Helly family. We describe characterizations for biclique-Helly graphs, leading to polynomial time recognition algorithms. In addition, we relate biclique-Helly graphs to the classes of clique-Helly, disk-Helly and neighborhood-Helly graphs.

Equiprojective polyhedra

A convex polyhedron P is equiprojective if, for some k, the orthogonal projection (or “shadow”) of P in every direction, except those directions parallel to faces of P, is a k-gon. We address an open question posed by Shepherd and reported in Croft, Falconer, and Guy's “Unsolved Problems in Geometry”, by characterizing equiprojective polyhedra, and giving an O ( n log n ) -time recognition algorithm.

Recognizing edge clique graphs among interval graphs and probe interval graphs

The edge clique graph of a graph H is the one having the edge set of H as vertex set, two vertices being adjacent if and only if the corresponding edges belong to a common complete subgraph of H . We characterize the graph classes {edge clique graphs} ∩ {interval graphs} as well as {edge clique graphs} ∩ {probe interval graphs}, which leads to polynomial time recognition algorithms for them. This work generalizes corresponding results in [M.R. Cerioli, J.L. Szwarcfiter, Edge clique graphs and some classes of chordal graphs, Discrete Math. 242 (2002) 31–39].

Probe split graphs

Graphs and Algorithms An undirected graph G=(V,E) is a probe split graph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and there exists E' ⊆ N× N such that G'=(V,E∪ E') is a split graph. Recently Chang et al. gave an O(V4(V+E)) time recognition algorithm for probe split graphs. In this article we give O(V2+VE) time recognition algorithms and characterisations by forbidden induced subgraphs both for the case when the partition into probes and non-probes is given, and when it is not given.

Recognizing Chordal Probe Graphs and Cycle-Bicolorable Graphs

A graph $G=(V,E)$ is a chordal probe graph if its vertices can be partitioned into two sets, $P$ (probes) and $N$ (non-probes), where $N$ is a stable set and such that $G$ can be extended to a chordal graph by adding edges between non-probes. We give several characterizations of chordal probe graphs, first, in the case of a fixed given partition of the vertices into probes and non-probes, and second, in the more general case where no partition is given. In both of these cases, our results are obtained by introducing new classes, namely, $N$-triangulatable graphs and cycle-bicolorable graphs. We give polynomial time recognition algorithms for each class. $N$-triangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. For cycle-bicolorable graphs, which are shown to be perfect, we prove that any cycle-bicoloring of a graph renders it $N$-triangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and non-probes, is $O(|P||E|)$, thus also providing an interesting tractable subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is $O(|E|^2)$.

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, Hüffner, 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.

GENERALIZING MONOTONICITY: ON RECOGNIZING SPECIAL CLASSES OF POLYGONS AND POLYHEDRA

A simple polyhedron is weakly-monotonic in direction [Formula: see text] provided that the intersection of the polyhedron and any plane with normal [Formula: see text] is simply-connected (i.e. empty, a point, a line-segment or a simple polygon). Furthermore, if the intersection is a convex set, then the polyhedron is said to be weakly-monotonic in the convex sense. Toussaint10 introduced these types of polyhedra as generalizations of the 2-dimensional notion of monotonicity. We study the following recognition problems: Given a simple n-vertex polyhedron in 3-dimensions, we present an O(n log n) time algorithm to determine if there exists a direction [Formula: see text] such that when sweeping over the polyhedron with a plane in direction [Formula: see text], the cross-section (or intersection) is a convex set. If we allow multiple convex polygons in the cross-section as opposed to a single convex polygon, then we provide a linear-time recognition algorithm. For simply-connected cross-sections (i.e. the cross-section is empty, a point, a line-segment or a simple polygon), we derive an O(n2) time recognition algorithm to determine if a direction [Formula: see text] exists. We then study variations of monotonicity in 2-dimensions, i.e. for simple polygons. Given a simple n-vertex polygon P, we can determine whether or not a line ℓ can be swept over P in a continuous manner but with varying direction, such that any position of ℓ intersects P in at most two edges. We study two variants of the problem: one where the line is allowed to sweep over a portion of the polygon multiple times and one where it can sweep any portion of the polygon only once. Although the latter problem is slightly more complicated than the former since the line movements are restricted, our solutions to both problems run in O(n2) time.

On algorithms for ( [formula omitted],gem)-free graphs

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. We present O ( n 2 ) time recognition algorithms for chordal gem-free graphs and for ( P 5 ,gem)-free graphs. Using a characterization of ( P 5 ,gem)-free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees [A. Brandstädt, D. Kratsch, On the structure of ( P 5 ,gem)-free graphs, Discrete Appl. Math. 145 (2005), 155–166], we give linear time algorithms for the following NP-complete problems on ( P 5 ,gem)-free graphs: Minimum Coloring; Maximum Weight Stable Set; Maximum Weight Clique; and Minimum Clique Cover.

Characterization and recognition of Helly circular-arc clique-perfect graphs

Abstract A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem. In this work we characterize clique-perfect graphs by a restricted list of minimal forbidden induced subgraphs when the graph is a Helly circular-arc graph. This characterization leads to a polynomial time recognition algorithm for clique-perfect graphs inside this class of graphs.