A linear time algorithm for computing a minimum paired-dominating set of a convex bipartite graph

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y is a convex bipartite graph, if there is an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ is consecutive with respect to the ordering of X, and G is said to have convexity with respect to X. A k-star caterpillar is a tree with a collection of stars with each star having k vertices of degree one whose roots are joined by a path. For a bipartite graph with partitions X and Y, we associate a k-star caterpillar on X such that for each vertex in Y, its neighborhood induces a tree. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph $$G=(V,E)$$ and a subset of vertices $$R \subseteq V(G)$$ , the objective is to find a minimum cardinality set $$S \subseteq V(G)$$ such that the set $$R \cup S$$ induces a connected subgraph. In this paper, we present the following dichotomy result: we show that STREE is NP-complete for 1-star caterpillar convex bipartite graphs and polynomial-time solvable for 0-star caterpillar convex bipartite graphs (also known as convex bipartite graphs). We also strengthen the well-known result of Müller and Brandstädt (Theoret Comput Sci 53(2-3):257-265, 1987), which says STREE in chordal bipartite graphs is NP-complete (reduction instances are 3-star caterpillar convex bipartite graphs). As an application, we use our STREE results to solve: (i) the classical dominating set problem in convex bipartite graphs, (ii) STREE on interval graphs, linear time.

The article considers symmetric linear spaces of bipartite graphs (SLSBG), i.e. the set of bipartite graphs with fixed lobes closed with respect to the symmetric difference and permutations of vertices in each lobe. The operation of symmetric difference itself is introduced in this work. The paper provides a structural description of all SLSBG. Symmetric linear spaces of bipartite graphs are divided into trivial (four SLSBG) and nontrivial. Non-trivial ones, in turn, are divided into two families. The first is C -series consisting only of bicomplete graphs, i.e. graphs that are a disjunct union of two complete bipartite graphs graph wings). The second family is D -series that includes graphs in which the degrees of vertices in one lobe have the same parity, and in the other lobe these degrees may be arbitrary. It is proved that every SLSBG of the D -series coincides with one of nine sets defined by the parity of the vertices’ degrees. For the SLSBG of the C -series it is obtained that every two-sided SLSBG (i.e., containing graphs whose both wings have nonempty lobes) is the intersection of the set of all bicomplete graphs with the set of all graphs with an even number of edges or with any space of the D -series.

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.

Dominating induced matching in some subclasses of bipartite graphs

A linear algorithm for the domination number of a series-parallel graph

A randomized polynomial time algorithm is presented which, for every simple, connected, k-regular graph on n vertices, finds a tour that visits every vertex and has length at most (1 + √(64/1n k)) n with high probability. The proof follows simply from results developed in the context of permanents; Egorychev's and Falikman's theorem which lower bounds the permanent of a doubly stochastic matrix and the polynomial time algorithm of Jerrum, Sinclair and Vigoda which samples a near-random, perfect matching from a bipartite graph. The techniques in this paper suggest new permanent-based approaches for TSP which could be useful in attacking other interesting cases of TSP.

Locating and paired-dominating sets in graphs

A note on internally disjoint alternating paths in bipartite graphs

Bipartite graphs are often used to illustrate relationships between two sets of data, such as web pages and visitors. At the same time, information is often organized hierarchically, for example, web pages are divided into directories by their contents. The hierarchical structures are useful for analyzing information. Graphs with both a bipartite structure and a hierarchical structure are called “clustered bipartite graphs.” A new clustered bipartite graphs visualization technique was developed for representing both bipartite and hierarchical structures simultaneously. In this technique, nodes in one set of the bipartite graph, which are leaves of a tree, are arranged in hierarchical multi-circular style. Then, nodes in the other set of the bipartite graph are arranged by the force-directed method. The technique enables step-by-step exploration for large-scale bipartite graphs.

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

A defect set in a bipartite graph with vertex classes V and W is a subset X ⊂ V such that the neighbourhood N(X) satisfies |N(X)| < |X|. We study a lemma on defect sets in bipartite graphs with certain expanding properties from the algorithmic complexity point of view. This lemma is the core of a result of Friedman and Pippenger which states that expanding graphs contain all small trees. We also discuss related problems of finding shortest circuits of matroids represented over a field. In particular, we propose a new straightforward method to derive a weaker form (PR-completeness) of the recent NP-completeness results of Khachiyan [11] and Vardy [18] concerning this problem for the field of rationals and GF(p m), respectively.KeywordsPolynomial TimeDecision ProblemBipartite GraphShort CircuitFinite FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the NP-completeness of the perfect matching free subgraph problem

Computing a minimum outer-connected dominating set for the class of chordal graphs

Complete Multipartite Graphs and their Null Set