A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

Given an undirected and vertex weighted graph G, the Weighted Feedback Vertex Problem (WFVP) consists in finding a subset F⊆V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve WFVP on it.     

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 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.     

On an infinite family of graphs with information ratio 2 − 1/k

In this paper we consider the secret sharing problem on special access structures with minimal qualified subsets of size two, i.e. secret sharing on graphs. This means that the participants are the vertices of the graph and the qualified subsets are the subsets of V(G) spanning at least one edge. The information ratio of a graph G is denoted by R(G) and is defined as the ratio of the greatest size of the shares a vertex has to remember and of the size of the secret. Since the determination of the exact information ratio is a non-trivial problem even for small graphs (i.e. for V(G) = 6), every construction can be of particular interest. Let k be the maximal degree in G. In this paper we prove that R(G) = 2 ? 1/k for every graph G with the following properties: (A) every vertex has at most one neighbour of degree one; (B) vertices of degree at least 3 are not connected by an edge; (C) the girth of the graph is at least 6. We prove this by using polyhedral combinatorics arguments and the entropy method.     

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.     

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.     

A Cubic Kernel for Feedback Vertex Set and Loop Cutset

The Feedback Vertex Set problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192–202, 2006), we show that this problem has a kernel with O(k ³) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G′ with O(k ³) vertices and integer k′≤k, such that G has a feedback vertex set of size at most k, if and only if G′ has a feedback vertex set of size at most k′. Moreover, the algorithm can be made constructive: if the reduced instance G′ has a feedback vertex set of size k′, then we can easily transform a minimum size feedback vertex set of G′ into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set.     

Two algorithms for minimum 2-connected r-hop dominating set

For a graph G=(V,E), a subset D⊆V is an r-hop dominating set if every vertex u∈V−D is at most r-hops away from D. It is a 2-connected r-hop dominating set if the subgraph of G induced by D is 2-connected. In this paper, we present two approximation algorithms to compute minimum 2-connected r-hop dominating set. The first one is a greedy algorithm using ear decomposition of 2-connected graphs. This algorithm is applicable to any 2-connected general graph. The second one is a three-phase algorithm which is only applicable to unit disk graphs. For both algorithms, performance ratios are given.     

Enumerating Minimal Subset Feedback Vertex Sets

The Subset Feedback Vertex Set problem takes as input a pair (G,S), where G=(V,E) is a graph with weights on its vertices, and S?V. The task is to find a set of vertices of total minimum weight to be removed from G, such that in the remaining graph no cycle contains a vertex of S. We show that this problem can be solved in time O(1.8638 ⁿ ), where n=|V|. This is a consequence of the main result of this paper, namely that all minimal subset feedback vertex sets of a graph can be enumerated in time O(1.8638 ⁿ ).     

k-tuple domination in graphs

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the k-tuple domination problem in strongly chordal graphs, which is a subclass of chordal graphs and includes trees, block graphs, interval graphs and directed path graphs. We also prove that the k-tuple domination problem is NP-complete for split graphs (a subclass of chordal graphs) and for bipartite graphs.     

On Finding Min-Min Disjoint Paths

The Min-Min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-hard and admits no K-approximation for any K>1 in the general case (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006). In this paper, we first show that Bhatia et al.’s NP-hardness proof (Bhatia et al. in J. Comb. Optim. 12:83–96, 2006), a claim of correction to Xu et al.’s proof (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006), for the edge-disjoint Min-Min problem in the general undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in the proof of Bhatia et al. (J. Comb. Optim. 12:83–96, 2006). We then gave a correct proof of NP-hardness of this problem in undirected graphs. Finally we give a polynomial-time algorithm for the vertex disjoint Min-Min problem in planar graphs by showing that the vertex disjoint Min-Min problem is polynomially solvable in st-planar graph G=(V,E) whose corresponding auxiliary graph G(V,E∪{e(st)}) can be embedded into a plane, and a planar graph can be decomposed into several st-planar graphs whose Min-Min paths collectively contain a Min-Min disjoint-path pair between s and t in the original graph G. To the best of our knowledge, these are the first polynomial algorithms for the Min-Min problems in planar graphs.     

A 2-approximation algorithm for the vertex cover P4 problem in cubic graphs

A subset F of vertices of a graph G is called a vertex cover P_k set if every path of order k in G contains at least one vertex from F. Denote by ψ_k(G) the minimum cardinality of a vertex cover P_k set in G. The vertex cover P_k (VCP_k) problem is to find a minimum vertex cover P_k set. It is easy to see that the VCP₂ problem corresponds to the well-known vertex cover problem. In this paper, we restrict our attention to the VCP₄ problem in cubic graphs. The paper proves that the VCP₄ problem is NP-hard for cubic graphs. Further, we give sharp lower and upper bounds on ψ₄(G) for cubic graphs and propose a 2-approximation algorithm for the VCP₄ problem in cubic graphs.     

An Exact Exponential Time Algorithm for Power Dominating Set

The Power Dominating Set problem is an extension of the well-known domination problem on graphs in a way that we enrich it by a second propagation rule: given a graph G(V,E), a set P?V is a power dominating set if every vertex is observed after the exhaustive application of the following two rules. First, a vertex is observed if v∈P or it has a neighbor in P. Secondly, if an observed vertex has exactly one unobserved neighbor u, then also u will be observed, as well. We show that Power Dominating Set remains $\mathcal{NP}$ -hard on cubic graphs. We design an algorithm solving this problem in time $\mathcal{O}^{*}(1.7548^{n})$ on general graphs, using polynomial space only. To achieve this, we introduce so-called reference search trees that can be seen as a compact representation of usual search trees, providing non-local pointers in order to indicate pruned subtrees.     

Oriented colorings of partial 2-trees

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping f from V(G) to V(H), that is f(x)f(y) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H. In this paper, we determine the oriented chromatic number of the class of partial 2-trees for every girth g?3. We also give an upper bound for the oriented chromatic number of planar graphs with girth at least 11.     

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O^∗(²n−z), where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O^∗(²n/2). Another implication is an algorithm for general graphs whose running time is O(n^1.7088).     

The on-line first-fit algorithm for radio frequency assignment problems

The radio frequency assignment problem is to minimize the number of frequencies used by transmitters with no interference in radio communication networks; it can be modeled as the minimum vertex coloring problem on unit disk graphs. In this paper, we consider the on-line first-fit algorithm for the problem and show that the competitive ratio of the algorithm for the unit disk graph G with χ(G)=2 is 3, where χ(G) is the chromatic number of G. Moreover, the competitive ratio of the algorithm for the unit disk graph G with χ(G)>2 is at least 4−3/χ(G). The average performance for the algorithm is also discussed in this paper.     

Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems

In the Hitting Set problem, we are given a collection F of subsets of a ground set V and an integer p, and asked whether V has a p-element subset that intersects each set in F. We consider two parameterizations of Hitting Set below tight upper bounds, p=m−k and p=n−k. In both cases k is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP ⊆ NP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph H=(V,F), makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of H=(V,F) is the minimum integer d such that for each X⊂V the hypergraph with vertex set V?X and edge set containing all edges of F without vertices in X, has a vertex of degree at most d.In Nonblocker (Directed Nonblocker), we are given an undirected graph (a directed graph) G on n vertices and an integer k, and asked whether G has a set X of n−k vertices such that for each vertex y∉X there is an edge (arc) from a vertex in X to y. Nonblocker can be viewed as a special case of Directed Nonblocker (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that Nonblocker has a linear-order kernel. We obtain a linear-order kernel for Directed Nonblocker.     

Improved Approximation Algorithms for the Spanning Star Forest Problem

A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set. We present a 0.71-approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. (SIAM J. Comput. 38:946–962, 2008). We also present a 0.64-approximation algorithm for the problem on node-weighted graphs. Finally, we present improved hardness of approximation results for the weighted (both edge-weighted and node-weighted) versions of the problem. Our algorithms use a non-linear rounding scheme, which might be of independent interest.     

Combinatorial algorithms for feedback problems in directed graphs

Given a weighted directed graph G=(V,A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A′⊆A such that the directed graph (V,A?A′) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple combinatorial algorithms for these problems that achieve an approximation ratio bounded by the length, in terms of number of arcs, of a longest simple cycle of the digraph.