Bipartite Distance-hereditary Graphs Research Articles

A tight relation between series-parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum value.

Computing a metric basis of a bipartite distance-hereditary graph

A vertex x of a connected graph G resolves two distinct vertices u and v in V(G) if the distance between u and x differs from the distance between v and x. A subset X of V(G)resolves two distinct vertices u and v in G if there exists a vertex x in X that resolves u and v; X is a metric generator of G if, for every pair of distinct vertices u and v of G, X resolves u and v and is a metric basis of G if X is a metric generator of G with minimum cardinality. The metric dimension of G is the cardinality of a metric basis of G. The problem of finding the metric dimension of an arbitrary graph is NP-hard. In this paper we show that the problem is solvable in linear time for the class of bipartite distance-hereditary graphs.

On the complexity of minimum maximal uniquely restricted matching

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. A matching M in a graph G is called a uniquely restricted matching if, G[V(M)], the subgraph of G induced by the set of M-saturated vertices of G contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any uniquely restricted matching of G. Given a graph G, the Min-Max-UR Matching problem asks to find a maximal uniquely restricted matching in G of minimum cardinality and Decide-Min-Max-UR Matching problem, the decision version of this problem takes a graph G and an integer k and asks whether G admits a maximal uniquely restricted matching of cardinality at most k. It is known that the Decide-Min-Max-UR Matching problem is NP-complete. In this paper, we strengthen this result by proving that the Decide-Min-Max-UR Matching problem remains NP-complete for chordal bipartite graphs, star-convex bipartite graphs, chordal graphs, and doubly chordal graphs. On the positive side, we prove that the Min-Max-UR Matching problem is polynomial time solvable for bipartite distance-hereditary graphs and linear time solvable for bipartite permutation graphs, proper interval graphs, and threshold graphs. Finally, we prove that the Min-Max-UR Matching problem is APX-complete for graphs with maximum degree 4.

The complexity of total k-domatic partition and total R-domination on graphs with weak elimination orderings

In this paper, we propose two linear-time algorithms. One is for computing a weak elimination ordering of a bipartite distance-hereditary graph, and the other one is an alternative algorithm to solve the total R-domination problem for any chordal bipartite graph with a weak elimination ordering. Our two linear-time algorithms lead to a unified approach to several variations of total domination problems for bipartite distance-hereditary graphs. We also show that tthe total 3-domatic partition problem is NP-complete for planar graphs of maximum degree 9 and planar bipartite graphs of maximum degree 12, and show that the 4-domatic partition problem for planar graphs of maximum degree d is polynomial-time reducible to the total 4-domatic partition problem for planar graphs of maximum degree d + 1.

On the geodetic iteration number of a graph in which geodesic and monophonic convexities are equivalent

Let G be a graph, u and v two vertices of G, and X a subset of V(G). A u-vgeodesic is a path between u and v of minimum length. Ig(u,v) is the set of vertices that lie on any u-v geodesic and Ig(X) is the set ⋃u,v∈XIg(u,v). X is g-convex if Ig(X)=X. Analogously, Im(u,v) is the set of vertices that lie on any induced path between u and v and Im(X) is the set ⋃u,v∈XIm(u,v). X is m-convex if Im(X)=X.The g-convex hull[X]g of X is the smallest g-convex set containing X. Igh(X) equals Ig(X), if h=1, and equals I(Igh−1(X)), if h>1. The geodetic iteration number, gin(X), of X in G is the smallest h such that Igh(X)=Igh+1(X)=[X]g. The geodetic iteration number of G, denoted by gin(G), is defined as gin(G)=max{gin(X)|X⊆V(G)}.In this paper we provide an O(n3m) time algorithm (where n and m are the cardinalities of the vertex set and of the edge set of the graph, respectively) to compute the geodetic iteration number of a graph belonging to the class, say Γ, of graphs in which the families of g-convex sets and of m-convex sets coincide (i.e., every g-convex set is m-convex). Since Γ properly contains the class of distance-hereditary graphs, this result extends the result in Dourado et al. (2016). Furthermore, we provide an O(n2m) time algorithm to compute the geodetic iteration number of a bipartite distance-hereditary graph.

Computing a metric basis of a 2-connected bipartite distance-hereditary graph

A vertex x of a connected graph G resolves two distinct vertices u and v in V(G) if the distance between u and x differs from the distance between v and x. A subset X of V(G)resolves two distinct vertices u and v in G if there exists a vertex x in X that resolves u and v; X is a metric generator of G if, for every pair of distinct vertices u and v of G, X resolves u and v and is a metric basis of G if X is a metric generator of G with minimum cardinality. The metric dimension of G is the cardinality of a metric basis of G. The problem of finding the metric dimension of an arbitrary graph is NP-hard. In this paper we show that the problem is solvable in polynomial time for the class of 2-connected bipartite distance-hereditary graphs by providing an algorithm that computes a metric basis of a 2-connected bipartite distance-hereditary graph in O(|V(G)|2|E(G)|) time.

Complexity of k-tuple total and total {k}-dominations for some subclasses of bipartite graphs

We consider two variations of graph total domination, namely, k-tuple total domination and total {k}-domination (for a fixed positive integer k). Their related decision problems are both NP-complete even for bipartite graphs. In this work, we study some subclasses of bipartite graphs. We prove the NP-completeness of both problems (for every fixed k) for bipartite planar graphs and we provide an APX-hardness result for the total domination problem for bipartite subcubic graphs. In addition, we introduce a more general variation of total domination (total (r,m)-domination) that allows us to design a specific linear time algorithm for bipartite distance-hereditary graphs. In particular, it returns a minimum weight total {k}-dominating function for bipartite distance-hereditary graphs.

Two classes of graphs in which some problems related to convexity are efficiently solvable

Monophonic, geodesic and 2-geodesic convexities ([Formula: see text]-convexity, [Formula: see text]-convexity and [Formula: see text]-convexity, for short) on graphs are based on the families of induced paths, shortest paths and shortest paths of length [Formula: see text], respectively. We introduce a class of graphs, the class of cross-cyclicgraphs, in which every connected [Formula: see text]-convex set is also [Formula: see text]-convex and [Formula: see text]-convex. We show that this class is properly contained in the class, say [Formula: see text], of graphs in which geodesic and monophonic convexities are equivalent and properly contains the class of distance-hereditary graphs. Moreover, we show that (1) an [Formula: see text]-hull set (i.e., a subset of vertices, with minimum cardinality, whose [Formula: see text]-convex hull equals the whole vertex set) and, hence, the m-hull number and the g-hull number of a graph in [Formula: see text] can be computed in polynomial time and that (2) both the geodesic-convex hull and the monophonic-convex hull can be computed in linear time in a cross-cyclic graph without cycles of length [Formula: see text] and, hence, in a bipartite distance-hereditary graph.

Total dominating sequences in trees, split graphs, and under modular decomposition

A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgrt(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgrt(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgrt(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgrt(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.

On the Galois lattice of bipartite distance hereditary graphs

We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a bipartite distance hereditary graph. Relations with the class of Ptolemaic graphs are discussed and exploited to give an alternative proof of the result.

Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

Graph Theory Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.

Recognition of probe distance-hereditary graphs

Let G denote a graph class. An undirected graph G is called a probe G graph if one can make G a graph in G by adding edges between vertices in some independent set of G. By definition graph class G is a subclass of probe G graphs. A graph is distance hereditary if the distance between any two vertices remains the same in every connected induced subgraph. Bipartite distance-hereditary graphs are both bipartite and distance hereditary. In this paper we propose O(nm)-time algorithms to recognize probe distance-hereditary graphs and probe bipartite distance-hereditary graphs where n and m are the numbers of vertices and edges of the input graph respectively.

Distance Hereditary Graphs and the Interlace Polynomial

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].We define the γ invariant as the coefficient of x1in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1whenntrue twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs

A biclique cover (resp. biclique decomposition) of a bipartite graph B is a family of complete bipartite subgraphs of B whose edges cover (resp. partition) the edges of B. The minimum cardinality of a biclique cover (resp. biclique decomposition) is denoted by s-dim( B) (resp. s-part( B)). The decision problems associated with the computation of s-dim and s-part are NP-complete for general bipartite graphs; the decision problem associated to s-dim is NP-complete for bipartite chordal graphs, and polynomial for bipartite distance-hereditary graphs, for bipartite convex graphs and for bipartite C4-free graphs. We show here that for bipartite domino-free graphs (a strict generalization of bipartite distance-hereditary graphs and bipartite C4-free graphs), s-dim and s-part are equal and can be computed in O(n × m) time. Moreover, we propose a O(n × m) time algorithm to check the domino-free property and to build the Galois lattice of such graphs.

THE PATH-PARTITION PROBLEM IN BIPARTITE DISTANCE-HEREDITARY GRAPHS

A path partition of a graph is a collection of vertex-disjoint paths that cover all vertices of the graph. The path-partition problem is to nd a path partition of minimum size. This paper gives a linear-time algorithm for the path-partition problem in bipartite distance-hereditary graphs.

THE PATH-PARTITION PROBLEM IN BIPARTITE DISTANCE-HEREDITARY GRAPHS

A path partition of a graph is a collection of vertex-disjoint paths that cover all vertices of the graph. The path-partition problem is to nd a path partition of minimum size. This paper gives a linear-time algorithm for the path-partition problem in bipartite distance-hereditary graphs.

Polynomial time algorithms for Hamiltonian problems on bipartite distance-hereditary graphs

By investigating the neighbourhood of the last pendant vertex in a one vertex extension-sequence we develop polynomial time algorithms for the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH in (6,2)-chordal bipartite graphs.