Minimal proper interval completions

Given an arbitrary graph G=(V,E) and a proper interval graph H=(V,F) with E⊆F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any sandwich graph H^′=(V,F^′) with E⊆F^′⊂F, H^′ is not a proper interval graph. In this paper we give a O(n+m) time algorithm computing a minimal proper interval completion of an arbitrary graph. The output is a proper interval model of the completion.     

Signed and minus clique-transversal functions on graphs

A minus (respectively, signed) clique-transversal function of a graph G=(V,E) is a function (respectively, {−1,1}) such that _∑u∈Cf(u)?1 for every maximal clique C of G. The weight of a minus (respectively, signed) clique-transversal function of G is f(V)=_∑v∈Vf(v). The minus (respectively, signed) clique-transversal problem is to find a minus (respectively, signed) clique-transversal function of G of minimum weight. In this paper, we present a unified approach to these two problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. We also prove that the signed clique-transversal problem is NP-complete for chordal graphs and planar graphs.     

Feedback vertex sets in star graphs

In a graph G=(V,E), a subset F⊂V(G) is a feedback vertex set of G if the subgraph induced by V(G)?F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.     

A linear-time algorithm for paired-domination problem in strongly chordal graphs

Let G=(V,E) be a simple graph without isolated vertices. A vertex set S⊆V is a paired-dominating set if every vertex in V−S has at least one neighbor in S and the induced subgraph G[S] has a perfect matching. In this paper, we present a linear-time algorithm to find a minimum paired-dominating set in strongly chordal graphs if the strong (elimination) ordering of the graph is given in advance.     

Locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs

A spanning tree T of a graph G=(V,E) is called a locally connected spanning tree if the set of all neighbors of v in T induces a connected subgraph of G for all v∈V. The problem of recognizing whether a graph admits a locally connected spanning tree is known to be NP-complete even when the input graphs are restricted to chordal graphs. In this paper, we propose linear time algorithms for finding locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs, respectively.     

The algorithmic complexity of mixed domination in graphs

Let G=(V,E) be a simple graph with vertex set V and edge set E. A subset W⊆V∪E is a mixed dominating set if every element x∈(V∪E)?W is either adjacent or incident to an element of W. The mixed domination problem is to find a minimum mixed dominating set of G. In this paper we first prove that a connected graph is a tree if and only if its total graph is strongly chordal, and thus we obtain a polynomial-time algorithm for this problem in trees. Further we design another linear-time labeling algorithm for this problem in trees. At the end of the paper, we show that the mixed domination problem is NP-complete even when restricted to split graphs, a subclass of chordal graphs.     

A note on finding all homogeneous set sandwiches

A module is a set of vertices H of a graph G=(V,E) such that each vertex of V?H is either adjacent to all vertices of H or to none of them. A homogeneous set is a nontrivial module. A graph G_s=(V,E_s) is a sandwich for a pair of graphs G_t=(V,E_t) and G=(V,E) if E_t⊆E_s⊆E. In a recent paper, Tang et al. [Inform. Process. Lett. 77 (2001) 17-22] described an O(Δn²) algorithm for testing the existence of a homogeneous set in sandwich graphs of G_t=(V,E_t) and G=(V,E) and then extended it to an enumerative algorithm computing all these possible homogeneous sets. In this paper, we invalidate this latter algorithm by proving there are possibly exponentially many such sets, even if we restrict our attention to strong modules. We then give a correct characterization of a homogeneous set of a sandwich graph.     

On vertex ranking of a starlike graph

A vertex coloring c:V→{1,2,…,t} of a graph G=(V,E) is a vertex t-ranking if for any two vertices of the same color every path between them contains a vertex of larger color. The vertex ranking number χ_r(G) is the smallest value of t such that G has a vertex t-ranking. A χ_r(G)-ranking of G is said to be an optimal vertex ranking. In this paper, we present an O(|V|+|E|) time algorithm for finding an optimal vertex ranking of a starlike graph G=(V,E). Our result implies that an optimal vertex ranking of a split graph can be computed in linear time.     

Simplicial powers of graphs

In a graph, a vertex is simplicial if its neighborhood is a clique. For an integer k≥1, a graph G=(V_G,E_G) is the k-simplicial power of a graph H=(V_H,E_H) (H a root graph of G) if V_G is the set of all simplicial vertices of H, and for all distinct vertices x and y in V_G, xyE_G if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k≤5, k-leaf powers can be recognized in linear time, and for k≤4, structural characterizations are known. For k≥6, the recognition and characterization problems of k-leaf powers are still open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs.     

Note on maximal bisection above tight lower bound

In a graph G=(V,E), a bisection (X,Y) is a partition of V into sets X and Y such that |X|?|Y|?|X|+1. The size of (X,Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G=(V,E) and are required to find a bisection of maximum size. It is not hard to see that ⌈|E|/2⌉ is a tight lower bound on the maximum size of a bisection of G.We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (Max-Bisec-ATLB): decide whether a graph G=(V,E) has a bisection of size at least ⌈|E|/2⌉+k, where k is the parameter. We show that this parameterized problem has a kernel with O(k²) vertices and O(k³) edges, i.e., every instance of Max-Bisec-ATLB is equivalent to an instance of Max-Bisec-ATLB on a graph with at most O(k²) vertices and O(k³) edges.     

On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms

For a graph G=(V,E) and a color set C, let f:E→C be an edge-coloring of G in which two adjacent edges may have the same color. Then, the graph G edge-colored by f is rainbow connected if every two vertices of G have a path in which all edges are assigned distinct colors. Chakraborty et al. defined the problem of determining whether the graph colored by a given edge-coloring is rainbow connected. Chen et al. introduced the vertex-coloring version of the problem as a variant, and we introduce the total-coloring version in this paper. We settle the precise computational complexities of all the three problems with regards to graph diameters, and also characterize these with regards to certain graph classes: cacti, outer planer and series-parallel graphs. We then give FPT algorithms for the three problems on general graphs when parameterized by the number of colors in C; our FPT algorithms imply that all the three problems can be solved in polynomial time for any graph with n vertices if |C|=O(logn).     

The differential of the strong product graphs

Let G=(V, E) be a graph of order n and let B(D) be the set of vertices in V ? D that have a neighbour in the set D. The differential of a set D is defined as ? (D)=|B(D)|?|D| and the differential of a graph to equal the maximum value of ?(D) for any subset D of V. In this paper we obtain several tight bounds for the differential of strong product graphs. In particular, we investigate the relationship between the differential of this type of product graphs and various parameters in the factors of the product.     

A Simpler and More Efficient Algorithm for the Next-to-Shortest Path Problem

Given an undirected graph G=(V,E) with positive edge lengths and two vertices s and t, the next-to-shortest path problem is to find an st-path which length is minimum amongst all st-paths strictly longer than the shortest path length. In this paper we show that the problem can be solved in linear time if the distances from s and t to all other vertices are given. Particularly our new algorithm runs in O(|V|log|V|+|E|) time for general graphs, which improves the previous result of O(|V|²) time and takes only linear time for unweighted graphs, planar graphs, and graphs with positive integer edge lengths.     

Finding Maximum Edge Bicliques in Convex Bipartite Graphs

A bipartite graph G=(A,B,E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v??A, vertices adjacent to v are consecutive in?B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G=(A,B,E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(nlog?³ nlog?log?n) time and O(n) space, where n=|A|. This improves the current O(n ²) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(n??(n)) and O(n) time, respectively, where n=min?(|A|,|B|) and ??(n) is the slowly growing inverse of the Ackermann function.     

The labeled perfect matching in bipartite graphs

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G=(V,E) with |V|=2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function , the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors.     

The edge-orientation problem and some of its variants on weighted graphs

Let G(V, E) be a connected undirected graph with n vertices and m edges, where each vertex v is associated with a cost C(v) and each edge e = (u, v) is associated with two weights, W(u → v) and W(v → u). The issue of assigning an orientation to each edge so that G becomes a directed graph is resolved in this paper. Determining a scheme to assign orientations of all edges such that max_x∈V{C(x)+∑_x→zW(x→z)} is minimized is the objective. This issue is called the edge-orientation problem (the EOP). Two variants of the EOP, the Out-Degree-EOP and the Vertex-Weighted EOP, are first proposed and then efficient algorithms for solving them on general graphs are designed. Ascertaining that the EOP is NP-hard on bipartite graphs and chordal graphs is the second result. Finally, an O(n log n)-time algorithm for the EOP on trees is designed. In general, the algorithmic results in this paper facilitate the implementation of the weighted fair queuing (WFQ) on real networks. The objective of the WFQ is to assign an effective weight for each flow to enhance link utilization. Our findings consequently can be easily extended to other classes of graphs, such as cactus graphs, block graphs, and interval graphs.     

Note on the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-unitary, proper subset H of a graph's vertices such that all vertices in H have the same neighbors outside H. Given two graphs G₁(V,E₁), G₂(V,E₂), the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph GS(V,ES), E₁⊆ES⊆E₂, which has a homogeneous set. Recently, Tang et al. [Inform. Process. Lett. 77 (2001) 17-22] proposed an interesting O(_?1⋅n²) algorithm for this problem, which has been considered its most efficient algorithm since. We show the incorrectness of their algorithm by presenting three counterexamples.     

Power Domination in Circular-Arc Graphs

A set S?V is a power dominating set (PDS) of a graph G=(V,E) if every vertex and every edge in G can be observed based on the observation rules of power system monitoring. The power domination problem involves minimizing the cardinality of a PDS of a graph. We consider this combinatorial optimization problem and present a linear time algorithm for finding the minimum PDS of an interval graph if the interval ordering of the graph is provided. In addition, we show that the algorithm, which runs in Θ(nlogn) time, where n is the number of intervals, is asymptotically optimal if the interval ordering is not given. We also show that the results hold for the class of circular-arc graphs.     

A good characterization of squares of strongly chordal split graphs

The square H² of a graph H is obtained from H by adding new edges between every two vertices having distance two in H. Lau and Corneil [Recognizing powers of proper interval, split and chordal graphs, SIAM J. Discrete Math. 18 (2004) 83-102] proved that recognizing squares of split graphs is an NP-complete problem. In contrast, we show that squares of strongly chordal split graphs can be recognized in quadratic-time by giving a structural characterization of these graph class.