Orderable Graphs Research Articles

Perfectly contractile graphs and quadratic toric rings

AbstractPerfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour, and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class of graphs that have no odd holes, no antiholes, and no odd stretchers as induced subgraphs. In particular, every graph belonging to is perfect. Everett and Reed conjectured that a graph belongs to if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph belongs to if and only if the toric ideal of the stable set polytope of is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.

Independence complexes of strongly orderable graphs

We prove that for any finite strongly orderable (generalized strongly chordal) graph G, the independence complex Ind(G) is either contractible or homotopy equivalent to a wedge of spheres of dimension at least bp(G)−1, where bp(G) is the biclique vertex partition number of G. In particular, we show that if G is a chordal bipartite graph, then Ind(G) is either contractible or homotopy equivalent to a sphere of dimension at least bp(G) − 1.

Efficient computation of the oriented chromatic number of recursively defined digraphs

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G=(V,E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

Computing a minimum paired-dominating set in strongly orderable graphs

A set D⊆V of a graph G=(V,E) is called a dominating set of G if every vertex in V∖D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if G[D], the subgraph of G induced by D, has a perfect matching. Given a graph G, Min-Paired-Dom-Set is the problem to find a paired-dominating set of G of minimum cardinality. Min-Paired-Dom-Set is known to be NP-hard for general graphs and many other restricted classes of graphs. However, Min-Paired-Dom-Set is solvable in polynomial time in some graph classes including strongly chordal graphs and chordal bipartite graphs. In this paper, we strengthen this result by proposing a polynomial time algorithm to compute a minimum paired-dominating set in the class of strongly orderable graphs, which includes strongly chordal graphs and chordal bipartite graphs. The algorithm runs in linear time if a quasi-simple elimination ordering of the strongly orderable graph is provided.

A Note on Self Complementary Brittle and Self Complementary Quasi Chordal Graphs

In this paper we deal with some classes of self-complementary (sc) perfectly orderable graphs namely sc brittle, sc quasi chordal graphs and propose algorithms for these classes. We obtain some results on these classes and an algorithm is proposed based on these results that recognize these classes. We also compile a catalogue for these classes up to 17 vertices.

An [formula omitted]-time certifying algorithm for recognizing HHD-free graphs

In this paper, we consider the recognition problem on a class of perfectly orderable graphs, namely, the HHD-free graphs; such graphs do not contain any induced subgraph isomorphic to a house, a hole, or a domino. We prove properties of the HHD-free graphs which enable us to present an O(nm)-time and O(n+m)-space algorithm for determining whether a graph on n vertices and m edges is HHD-free; currently, this is the fastest algorithm for this problem. We also describe how the algorithm can be augmented to provide a certificate (an induced house, hole, or domino) whenever it decides that the input graph is not HHD-free, thus answering an open question posed by Hoàng and Sritharan (Theoretical Computer Science 259 (2001) 233–244). The certificate computation requires O(n+m) additional time and O(n) space.

Formula omitted]-labeling of dually chordal graphs and strongly orderable graphs

An L(2,1)-labeling of a graph G=(V,E) is a function f:V(G)→{0,1,2,…} such that |f(u)−f(v)|⩾2 whenever uv∈E(G) and |f(u)−f(v)|⩾1 whenever u and v are at distance two apart. The span of an L(2,1)-labeling f of G, denoted as SP2(f,G), is the maximum value of f(x) over all x∈V(G). The L(2,1)-labeling number of a graph G, denoted as λ(G), is the least integer k such that G admits an L(2,1)-labeling of span k. The problem of computing λ(G) of a graph is known to be NP-complete. Griggs and Yeh have conjectured that λ(G)⩽Δ2(G) for a graph G with maximum degree, Δ(G), at least two. In this paper, we propose constant approximation algorithms for the problem of computing λ(G) for dually chordal graphs and strongly orderable graphs. As a by-product, we prove Griggs and Yeh Conjecture for dually chordal graphs and for those strongly orderable graphs whose maximum degrees are different from three. Finally, we propose a 2-approximation algorithm for computing λ(G) for chordal bipartite graphs, a special subclass of strongly orderable graphs, and prove that Griggs and Yeh Conjecture holds true for this class of graphs.

The Story of Perfectly Orderable Graphs

We give the story behind one examplar of the work of Vasek Chvatal, namely his conception of the class of perfectly orderable graphs.

An O(n3)-Time Recognition Algorithm for hhds-free Graphs

The class of hhds-free graphs properly generalizes the classes of strongly chordal graphs and distance-hereditary graphs, and forms a restriction of the class of perfectly orderable graphs. The problem of recognizing hhds-free graphs in polynomial time was posed by Brandstadt (Problem session, Dagstuhl seminor No. 04221, 2004), and Nikolopoulos and Palios (Proceedings of the 31st International Workshop on Graph Theoretic Concepts in Computer Science, 2005) gave an O(m n 2) algorithm. We present an O(n 3)-time algorithm for the problem. Our algorithm demonstrates a relationship between hhds-free graphs and strongly chordal graphs similar to that which is known to exist between hhd-free graphs and chordal graphs.

On compact and efficient routing in certain graph classes

In this paper we refine the notion of tree-decomposition by introducing acyclic ( R , D ) -clustering, where clusters are subsets of vertices of a graph and R and D are the maximum radius and the maximum diameter of these subsets. We design a routing scheme for graphs admitting induced acyclic ( R , D ) -clustering where the induced radius and the induced diameter of each cluster are at most 2. We show that, by constructing a family of special spanning trees, one can achieve a routing scheme of deviation Δ ⩽ 2 R with labels of size O ( log 3 n / log log n ) bits per vertex and O ( 1 ) routing protocol for these graphs. We investigate also some special graph classes admitting induced acyclic ( R , D ) -clustering with induced radius and diameter less than or equal to 2, namely, chordal bipartite, homogeneously orderable, and interval graphs. We achieve the deviation Δ = 1 for interval graphs and Δ = 2 for chordal bipartite and homogeneously orderable graphs.

On a conjecture of Hoàng and Tu concerning perfectly orderable graphs

Hoàng and Tu [On the perfect orderability of unions of two graphs, J. Graph Theory 33 (2000) 32–43] conjectured that a weakly triangulated graph which does not contain a chordless path with six vertices is perfectly orderable. We present a counter example to this conjecture.

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

In this paper, we consider the recognition problem on three classes of perfectly orderable graphs, namely, the HH-free, the HHD-free, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min \m α (n,n), m + n^2 log n\) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min \m α (n,n), m + n^2 log n\)-time and O(n+m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement øverlineG of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n^2) space, which leads to an O(n m)-time and O(n^2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n^3) time and O(n^2) space.

Linear Orderings of Subfamilies of AT‐Free Graphs

Asteroidal triple free (AT‐free) graphs have been introduced as a generalization of interval graphs, since interval graphs are exactly the chordal AT‐free graphs. While for interval graphs it is obvious that there is always a linear ordering of the vertices, such that for each triple of independent vertices the middle one intercepts any path between the remaining vertices of the triple, it is not clear that such an ordering exists for AT‐free graphs in general. In this paper we study graphs that are defined by enforcing such an ordering. In particular, we introduce two subfamilies of AT‐free graphs, namely, path orderable graphs and strong asteroid free graphs. Path orderable graphs are defined by a linear ordering of the vertices that is a natural generalization of the ordering that characterizes cocomparability graphs. On the other hand, motivation for the definition of strong asteroid free graphs comes from the fundamental work of Gallai on comparability graphs. We show that cocomparability graphs $\subset$ path orderable graphs $\subset$ strong asteroid free graphs $\subset$ AT‐free graphs. In addition, we settle the recognition question for the two new classes by proving that recognizing path orderable graphs is NP‐complete, whereas the recognition problem for strong asteroid free graphs can be solved in polynomial time.

Recognition of some perfectly orderable graph classes

This paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P 4-simplicial graphs are recognized in O( n 3.376) time, improving the previous bounds of O( n 4) and O( n 5), respectively. Brittle and semi-simplicial graphs are recognized in O( n 3) time using a randomized algorithm, and O(n 3 log 2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O( m 2). Welsh–Powell opposition graphs are recognized in O( n 3) time, improving the previous bound of O( n 4). HHP-free graphs and maxibrittle graphs are recognized in O( mn) and O( n 3.376) time, respectively.

New classes of perfectly orderable graphs

This paper generalizes previous works on perfectly orderable graphs by Olariu (Discrete Math. 113 (1992) 143) and by Hoàng et al. (Discrete Math. 102 (1992) 67). Chvátal defined a graph to be perfectly orderable (V. Chvátal, in: C. Berge, V. Chvátal (Eds.), Topics on Perfect Graphs, Annals of Discrete Mathematics, Vol. 21, North-Holland, Amsterdam, 1984, pp. 63–65) if there exists a linear order < on its set of vertices such that no induced path abcd with edges ab, bc, cd has both a< b and d< c. Given a graph G and a vertex v in G such that G− v is perfectly orderable, we set some conditions on v for which we deduce that G is perfectly orderable. Our method allows to construct a new class of such graphs, recognizable in polynomial time, containing quasi-brittle graphs, charming graphs and some other classes of perfectly orderable graphs.

Strongly orderable graphs A common generalization of strongly chordal and chordal bipartite graphs

In this paper those graphs are studied for which a so-called strong ordering of the vertex set exists. This class of graphs, called strongly orderable graphs, generalizes the strongly chordal graphs and the chordal bipartite graphs in a quite natural way. We consider two characteristic elimination orderings for strongly orderable graphs, one on the vertex set and the second on the edge set, and prove that these graphs can be recognized in O(| V|+| E|)| V| time. Moreover, a special strong ordering of a strongly orderable graph can be produced in the same time bound. We present variations of greedy algorithms that compute a minimum coloring, a maximum clique, a minimum clique partition and a maximum independent set of a strongly orderable graph in linear time if such a special strong ordering is given.

R-domination problems on homogeneously orderable graphs

In this paper, we consider r-dominating cliques in homogeneously orderable graphs (a common generalization of dually chordal and distance-hereditary graphs) and their relation to strict r-packing sets. We prove that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x, y of G d(x, y) ≤ r(x) + r(y) + 1 holds where r : V → N is a given vertex function. Furthermore, we show that for homogeneously orderable graphs with r-dominating cliques the cardinality of a maximum strict r-packing set equals the cardinality of a minimum r-dominating clique provided the last parameter is not two. Finally, we present two efficient algorithms: The first one decides whether a given homogeneously orderable graph has an r-dominating clique and, if so, computes both a minimum r-dominating clique and a maximum strict r-packing set of the graph. The second one computes a minimum connected r-dominating set in a homogeneously orderable graph. © 1997 John Wiley & Sons, Inc. Networks 30: 121–131, 1997

R‐domination problems on homogeneously orderable graphs

In this paper, we consider r-dominating cliques in homogeneously orderable graphs (a common generalization of dually chordal and distance-hereditary graphs) and their relation to strict r-packing sets. We prove that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x, y of G d(x, y) ≤ r(x) + r(y) + 1 holds where r : V → N is a given vertex function. Furthermore, we show that for homogeneously orderable graphs with r-dominating cliques the cardinality of a maximum strict r-packing set equals the cardinality of a minimum r-dominating clique provided the last parameter is not two. Finally, we present two efficient algorithms: The first one decides whether a given homogeneously orderable graph has an r-dominating clique and, if so, computes both a minimum r-dominating clique and a maximum strict r-packing set of the graph. The second one computes a minimum connected r-dominating set in a homogeneously orderable graph. © 1997 John Wiley & Sons, Inc. Networks 30: 121–131, 1997

Homogeneously orderable graphs

In this paper we introduce homogeneously orderable graphs which are a common generalization of distance-hereditary graphs, dually chordal graphs and homogeneous graphs. We present a characterization of the new class in terms of a tree structure of the closed neighborhoods of homogeneous sets in 2-graphs which is closely related to the defining elimination ordering. Moreover, we characterize the hereditary homogeneously orderable graphs by forbidden induced subgraphs as the house-hole-domino-sun-free graphs. The local structure of homogeneously orderable graphs implies a simple polynomial-time recognition algorithm for these graphs. Finally, we give a polynomial-time solution for the Steiner tree problem on homogeneously orderable graphs which extends the efficient solutions of that problem on distance-hereditary graphs, dually chordal graphs and homogeneous graphs.

Properly orderable graphs

In a graph G = ( V, E) provided with a linear order ‘ < ’ on V, a chordless path with vertices a, b, c, d and edges ab, bc, cd is called an obstruction if both a < b and d < c hold. Chvátal (1984) defined the class of perfectly orderable graphs (i.e., graphs possessing an acyclic orientation of the edges such that no obstruction is induced) and proved that they are perfect. We introduce here the class of properly orderable graphs which is a generalization of Chvátal's class of perfectly orderable graphs: obstructions are forbidden only in the subgraphs induced by the vertices of an odd cycle. We prove the perfection of these graphs and give an O( m 2 + mn + n) colouring algorithm.