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.     

Spanners of bounded degree graphs

A k-spanner of a graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most k times the distance in G. We prove that for fixed k,w, the problem of deciding if a given graph has a k-spanner of treewidth w is fixed-parameter tractable on graphs of bounded degree. In particular, this implies that finding a k-spanner that is a tree (a tree k-spanner) is fixed-parameter tractable on graphs of bounded degree. In contrast, we observe that if the graph has only one vertex of unbounded degree, then Treek-Spanner is NP-complete for k?4.     

Collective Tree Spanners in Graphs with Bounded Parameters

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system $\mathcal{T}(G)In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system T(G)\mathcal{T}(G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ? T(G)T\in\mathcal{T}(G) exists such that d _T (x,y)≤d _G (x,y)+r. We describe a general method for constructing a “small” system of collective additive tree r-spanners with small values of r for “well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O(?n)O(\sqrt{n}) collective additive tree 0-spanners, any weighted graph with tree-width at most k−1 admits a system of klog ₂ n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of klog _3/2 n collective additive tree (2w)(2\mathsf{w}) -spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log ₂ n collective additive tree (2?c/2?w)(2\lfloor c/2\rfloor\mathsf{w}) -spanners and a system of 4log ₂ n collective additive tree (2(?c/3?+1)w)(2(\lfloor c/3\rfloor +1)\mathsf {w}) -spanners (here, w\mathsf{w} is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log ₂ n collective additive tree (2w)(2\mathsf{w}) -spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.     

A Linear-Time Algorithm for Finding Locally Connected Spanning Trees on Circular-Arc Graphs

Suppose that T is a spanning tree of a graph G. T is called a locally connected spanning tree of G if for every vertex of T, the set of all its neighbors in T induces a connected subgraph of G. In this paper, given an intersection model of a circular-arc graph, an O(n)-time algorithm is proposed that can determine whether the circular-arc graph contains a locally connected spanning tree or not, and produce one if it exists.     

The super-connected property of recursive circulant graphs

In a graph G, a k-container C_k(u,v) is a set of k disjoint paths joining u and v. A k-container C_k(u,v) is k∗-container if every vertex of G is passed by some path in C_k(u,v). A graph G is k∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1?k?m. In this paper, we prove that the recursive circulant graphs G(2^m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35-62], are super-connected if and only if m≠2.     

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.     

On Clark and Suen bound-type results for k-domination and Roman domination of Cartesian product graphs

A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ _u∈V f(u). The Roman domination number γ_R(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ _k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γ_R(H)/3.     

Minimizing the makespan in a single machine scheduling problem with a time-based learning effect

Both the building cost and the multiple-source routing cost are important considerations in construction of a network system. A spanning tree with minimum building cost among all spanning trees is called a minimum spanning tree (MST), and a spanning tree with minimum k-source routing cost among all spanning trees is called a k-source minimum routing cost spanning tree (k-MRCT). This paper proposes an algorithm to construct a spanning tree T for a metric graph G with a source vertex set S such that the building cost of T is at most 1+2/(α−1) times of that of an MST of G, and the k-source routing cost of T is at most α(1+2(k−1)(n−2)/k(n+k−2)) times of that of a k-MRCT of G with respect to S, where α>1, k=|S| and n is the number of vertices of G.     

Parameterized Complexity of the Spanning Tree Congestion Problem

We study the problem of determining the spanning tree congestion of a?graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k??8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k??3 the problem becomes polynomially time solvable.     

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

A set of k spanning trees rooted at the same vertex r in a graph G is said to be independent if for each vertex x other than r, the k paths from r to x, one path in each spanning tree, are internally disjoint. Using independent spanning trees (ISTs) one can design fault-tolerant broadcasting schemes and increase message security in a network. Thus, the problem of ISTs on graphs has been received much attention. Recently, Yang et al. proposed a parallel algorithm for generating optimal ISTs on the hypercube. In this paper, we propose a similar algorithm for generating optimal ISTs on Cartesian product of complete graphs. The algorithm can be easily implemented in parallel or distributed systems. Moreover, the proof of its correctness is simpler than that of Yang et al.     

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 the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S ₁, S ₂, …, S _k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S _i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Lucky labelings of graphs

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u)≠S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,…,k} is the lucky number of G, denoted by η(G).Using algebraic methods we prove that η(G)?k+1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η(T)?2 for every tree T, and η(G)?3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that for every planar graph G. Nevertheless we offer a provocative conjecture that η(G)?χ(G) for every graph G.     

On the complexity of some colorful problems parameterized by treewidth

In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph.

1.

The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C_v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C_v, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.

2.

An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W[1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterized by the number of colors on the lists.

3.

The list chromatic numberχ_l(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L_v of colors, where each list has length at least r, there is a choice of one color from each vertex list L_v yielding a proper coloring of G. We show that the problem of determining whether χ_l(G)?r, the List Chromatic Number problem, is solvable in linear time on graphs of constant treewidth.

     

On the k-tuple domination of de Bruijn and Kautz digraphs

In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs.     

Editing to Eulerian graphs

The Eulerian Editing problem asks, given a graph G and an integer k, whether G can be modified into an Eulerian graph using at most k edge additions and edge deletions. We show that this problem is polynomial-time solvable for both undirected and directed graphs. We generalize these results for problems with degree parity constraints and degree balance constraints, respectively. We also consider the variants where vertex deletions are permitted. Combined with known results, this leads to full complexity classifications for both undirected and directed graphs and for every subset of the three graph operations.     

The k-in-a-Path Problem for Claw-free Graphs

The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s _i ,t _i ) for i=1,…,k and is to test whether G contain k mutually induced paths P _i such that P _i connects s _i and t _i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.     

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

Consider any undirected and simple graph G=(V, E), where V and E denote the vertex set and the edge set of G, respectively. Let |G|=|V|=n. The well-known Ore's theorem states that if deg_G(u)+deg_G(v)≥n+k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k=?1, hamiltonian for k=0, and hamiltonian-connected for k=1. Lin et al. generalized Ore's theorem and showed that under the same condition as above, G is r*-connected for 1≤r≤k+2 with k≥1. In this paper, we improve both theorems by showing that the hamiltonicity or r*-connectivity of any graph G satisfying the condition deg_G(u)+deg_G(v)≥n+k with k≥?1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.     

On the oriented chromatic number of grids

In this paper, we focus on the oriented coloring of graphs. Oriented coloring is a coloring of the vertices of an oriented graph G without symmetric arcs such that (i) no two neighbors in G are assigned the same color, and (ii) if two vertices u and v such that (u,v)∈A(G) are assigned colors c(u) and c(v), then for any (z,t)∈A(G), we cannot have simultaneously c(z)=c(v) and c(t)=c(u). The oriented chromatic number of an unoriented graph G is the smallest number k of colors for which any of the orientations of G can be colored with k colors.The main results we obtain in this paper are bounds on the oriented chromatic number of particular families of planar graphs, namely 2-dimensional grids, fat trees and fat fat trees.