HHD-free Graphs Research Articles

Walk Domination and HHD-Free Graphs

Walk Domination and HHD-Free Graphs

On the geodeticity of the contour of a graph

The eccentricity of a vertex v in a graph G is the maximum distance of v from any other vertex of G and v is a contour vertex of G if each vertex adjacent to v has eccentricity not greater than the eccentricity of v. The set of contour vertices of G is geodetic if every vertex of G lies on a shortest path between a pair of contour vertices. In this paper, we firstly investigate the existence of operations on graphs that allow to construct graphs in which the contour is geodetic. Then, after providing an alternative proof of the fact that the contour is geodetic in every HHD-free graph, we show that the contour is geodetic in every cactus and in every graph whose blocks are HHD-free or cycles or cographs. Finally, we generalize the above result by introducing the concept of geodetic-contour-preserving class of graphs and by proving that, if each block B in a graph G belongs to a class GB of graphs which is geodetic-contour-preserving, then the contour of G is geodetic.

Bipartite powers of k-chordal graphs

Graph Theory Let k be an integer and k ≥3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if Gm is chordal then so is Gm+2. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if Gm is k-chordal, then so is Gm+2. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m-th bipartite power G[m] of a bipartite graph G is the bipartite graph obtained from G by adding edges (u,v) where dG(u,v) is odd and less than or equal to m. Note that G[m] = G[m+1] for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G[m], where k, m are positive integers with k≥4.

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.

Maximum induced matching problem on hhd-free graphs

We show that the maximum induced matching problem can be solved on hhd-free graphs in O ( m 2 ) time; hhd-free graphs generalize chordal graphs and the previous best bound was O ( m 3 ) . Then, we consider a technique used by Brandstädt and Hoàng (2008) [4] to solve the problem on chordal graphs. Extending this, we show that for a subclass of hhd-free graphs that is more general than chordal graphs the problem can be solved in linear time. We also present examples to demonstrate the tightness of our results.

Finding a Sun in Building-Free Graphs

Deciding whether an arbitrary graph contains a sun was recently shown to be NP-complete (Hoang in SIAM J Discret Math 23:2156–2162, 2010). We show that whether a building-free graph contains a sun can be decided in O(min{mn 3, m 1.5 n 2}) time and, if a sun exists, it can be found in the same time bound. The class of building-free graphs contains many interesting classes of perfect graphs such as Meyniel graphs which, in turn, contains classes such as hhd-free graphs, i-triangulated graphs, and parity graphs. Moreover, there are imperfect graphs that are building-free. The class of building-free graphs generalizes several classes of graphs for which an efficient test for the presence of a sun is known. We also present a vertex elimination scheme for the class of (building, gem)-free graphs. The class of (building, gem)-free graphs is a generalization of the class of distance hereditary graphs and a restriction of the class of (building, sun)-free graphs.

All 4-Edge-Connected HHD-Free Graphs are $${\mathbb{Z}_3}$$ -Connected

An undirected graph G = (V, E) is called $${\mathbb{Z}_3}$$ -connected if for all $${b: V \rightarrow \mathbb{Z}_3}$$ with $${\sum_{v \in V}b(v)=0}$$, an orientation D = (V, A) of G has a $${\mathbb{Z}_3}$$ -valued nowhere-zero flow $${f: A\rightarrow \mathbb{Z}_3-\{0\}}$$ such that $${\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)}$$ for all $${v \in V}$$. We show that all 4-edge-connected HHD-free graphs are $${\mathbb{Z}_3}$$ -connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the $${\mathbb{Z}_3}$$ -connectivity for 4-edge-connected chordal graphs.

On the Complexity of Finding a Sun in a Graph

The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs (graphs containing no house, hole, or domino), have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun is NP-complete.

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.

Steiner distance and convexity in graphs

We use the Steiner distance to define a convexity in the vertex set of a graph, which has a nice behavior in the well-known class of HHD-free graphs. For this graph class, we prove that any Steiner tree of a vertex set is included into the geodesical convex hull of the set, which extends the well-known fact that the Euclidean convex hull contains at least one Steiner tree for any planar point set. We also characterize the graph class where Steiner convexity becomes a convex geometry, and provide a vertex set that allows us to rebuild any convex set, using convex hull operation, in any graph.

Finding a central vertex in an HHD-free graph

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 g...

Finding a central vertex in an HHD-free graph

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 five) as induced subgraphs. We present an algorithm which finds a central vertex of a HHD-free graph in O( Δ 1.376| V|) time, where Δ 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.

Finding houses and holes in graphs

A house is the complement of an induced path on five vertices. A hole is an induced cycle on five or more vertices. A domino is the cycle on six vertices with a long chord. A graph is HH-free if it does not contain a house or a hole. A graph is HHD-free if it does not contain a house, or a hole, or a domino. We present O(n3) algorithms to recognize HH-free graphs and HHD-free graphs. The previous best algorithms for the problems run in O(n4) time.

Convexity and HHD-Free Graphs

It is well known that chordal graphs can be characterized via m-convexity. In this paper we introduce the notion of m3 -convexity (a relaxation of m-convexity) which is closely related to semisimplicial ordering of graphs. We present new characterizations of HHD-free graphs via m3 -convexity and obtain some results known from [B. Jamison and S. Olariu, Adv. Appl. Math., 9 (1988), pp. 364--376] as corollaries. Moreover, we characterize weak bipolarizable graphs as the graphs for which the family of all m3 -convex sets is a convex geometry. As an application of our results we present a simple efficient criterion for deciding whether a HHD-free graph contains a r-dominating clique with respect to a given vertex radius function r.

Lexbfs-orderings and powers of hhd-free graphs *

The k-th power of a graph G is the graph with the same vertex set as G where two vertices are adjacent iff their distance in G is at most k. We investigate LexBFS-orderings of HHD-free graphsi.e., the graphs where each cycle of length at least five has two chords, with respect to their powers, It is well-known that any LexBFS-ordering of a HHD-free graph is a so-called semisimplicial ordering. We show, that any LexBFS-ordering of a HHD-free graph is a common semisimplicial ordering of all its odd powers. Furthermore, we characterize those HHD-free graphs by forbidden subgraphs for which any LexBFS-ordering of the initial graph is a common perfect elimination ordering of all odd or even powers

Powers of hhd-free graphs*

The k-the power of a graph G is the graph with the same vertex set as G where two vertices are adjacent iff their distance in G is at most k. In this paper we consider HHD-free graphs, i.e., the graphs where each cycle of length at least five has two chords. We show, that odd powers of HHD-free graphs are again HHD-free, and characterize those HHD-free graphs by forbidden subgraphs for which odd powers are chordal, and even powers are HHD-free or chordal, respectively.

Duchet-type theorems for powers of HHD-free graphs

Using an idea due to P. Duchet in proving his well-known theorem on powers of chordal graphs, we shall describe some theorems of Duchet-type for powers of graphs that have no long induced cycles. In particular, our Duchet-type theorem for HHD-free graphs improves a recent result due to Dragan, Nicolai and Brandstädt; stating that odd powers of HHD-free graphs are also HHD-free.

An nc algorithm to recognize hhd-free graphs

The HHD-free graphs are a class of perfect graphs that strictly contain the cographs, the chordal graphs, the Matula-perfect graphs, and the Welsh-Powell opposition graphs. In this paper we present two characterizations of the HHD-free graphs and show how to use them to produce a parallel recognition algorithm which runs in O(log n) time using O(n 4) processors on a concurrent-read, concurrent-write parallel random access machine.