Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs

For an ordered subset W= w₁, w₂,?…?w_k of vertices and a vertex u in a connected graph G, the representation of u with respect to W is the ordered k-tuple r(u|W)=(d(u, w₁), d(u, w₂),?…?, d(u, w_k)), where d(x, y) represents the distance between the vertices x and y. The set W is a local metric generator for G if every two adjacent vertices of G have distinct representations. A minimum local metric generator is called a local metric basis for G and its cardinality the local metric dimension of G. We show that the computation of the local metric dimension of a graph with cut vertices is reduced to the computation of the local metric dimension of the so-called primary subgraphs. The main results are applied to specific constructions including bouquets of graphs, rooted product graphs, corona product graphs, block graphs and chain of 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.     

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.     

Bandwidth on AT-free graphs

We study the classical Bandwidth problem from the viewpoint of parametrised algorithms. Given a graph G=(V,E) and a positive integer k, the Bandwidth problem asks whether there exists a bijective function β:{1,…,∣V∣}→V such that for every edge uv∈E, ∣β⁻¹(u)−β⁻¹(v)∣≤k. It is known that under standard complexity assumptions, no algorithm for Bandwidth with running time of the form f(k)n^O(1) exists, even when the input is restricted to trees. We initiate the search for classes of graphs where such algorithms do exist. We present an algorithm with running time n⋅2^O(klogk) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial algorithm that shows fixed-parameter tractability of Bandwidth on a graph class on which the problem remains NP-complete.     

The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs

We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T\mathcal{T} of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to T\mathcal{T} is a set of vertex-disjoint paths ℘ that covers the vertices of G such that the k vertices of T\mathcal{T} are all endpoints of the paths in ℘. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T\mathcal{T} is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49–64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.     

A General Approach to L（h,k）-Label Interconnection Networks

Given two non-negative integers h and k, an L（h, k）-labeling of a graph G = （V, E） is a function from the set V to a set of colors, such that adjacent nodes take colors at distance at least h, and nodes at distance 2 take colors at distance at least k. The aim of the L（h, k）-labeling problem is to minimize the greatest used color. Since the decisional version of this problem is NP-complete, it is important to investigate particular classes of graphs for which the problem can be efficiently solved. It is well known that the most common interconnection topologies, such as Butterfly-like, Beneg, CCC, Trivalent Cayley networks, are all characterized by a similar structure： they have nodes organized as a matrix and connections are divided into layers. So we naturally introduce a new class of graphs, called （l × n）-multistage graphs, containing the most common interconnection topologies, on which we study the L（h, k）-labeling. A general algorithm for L（h, k）-labeling these graphs is presented, and from this method an efficient L（2, 1）-labeling for Butterfly and CCC networks is derived. Finally we describe a possible generalization of our approach.     

Algorithms for (0, 1, d)-graphs with d constrains

Let G be a graph with vertex set V(G). Let n, k, d be non-negative integers such that n+2k+d≤|V(G)|?2 and |V(G)|?n?d are even. A matching which saturates exactly |V(G)|?d vertices is called a defect-d matching of G. If when deleting any n vertices the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect-d matching, then G is said to be an (n, k, d)-graph. We present an algorithm to determine (0, 1, d)-graphs with d constraints. Moreover, we solve the problem of augmenting a bipartite graph G=(B, W) to be a (0, 1, d)-graph by adding fewest edges, where d=∥B|?|W∥. The latter problem is applicable to the job assignment problem, where the number of jobs does not equal the number of persons.     

Exact exponential-time algorithms for finding bicliques

Due to a large number of applications, bicliques of graphs have been widely considered in the literature. This paper focuses on non-induced bicliques. Given a graph G=(V,E) on n vertices, a pair (X,Y), with X,Y⊆V, X∩Y=∅, is a non-induced biclique if {x,y}∈E for any x∈X and y∈Y. The NP-complete problem of finding a non-induced (k₁,k₂)-biclique asks to decide whether G contains a non-induced biclique (X,Y) such that |X|=k₁ and |Y|=k₂. In this paper, we design a polynomial-space O(n^1.6914)-time algorithm for this problem. It is based on an algorithm for bipartite graphs that runs in time O(n^1.30052). In deriving this algorithm, we also exhibit a relation to the spare allocation problem known from memory chip fabrication. As a byproduct, we show that the constraint bipartite vertex cover problem can be solved in time O(n^1.30052).     

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.     

General-demand disjoint path covers in a graph with faulty elements

A k-disjoint path cover of a graph is a set of k internally vertex-disjoint paths which cover the vertex set with k paths and each of which runs between a source and a sink. Given that each source and sink v is associated with an integer-valued demand d(v)≥1, we are concerned with general-demand k-disjoint path cover in which every source and sink v is contained in the d(v) paths. In this paper, we present a reduction of a general-demand disjoint path cover problem to an unpaired many-to-many disjoint path cover problem, and obtain some results on disjoint path covers of restricted HL-graphs and proper interval graphs with faulty vertices and/or edges.     

Path-matching problems

The notion of matching in graphs is generalized in this paper to a set of paths rather than to a set of edges. The generalized problem, which we call thepath-matching problem, is to pair the vertices of an undirected weighted graph such that the paths connecting each pair are subject to certain objectives and/or constraints. This paper concentrates on the case where the paths are required to be edge-disjoint and the objective is to minimize the maximal cost of a path in the matching (i.e., the bottleneck version). Other variations of the problem are also mentioned. Two algorithms are presented to find the best matching under the constraints listed above for trees. Their worst-case running times areO(n logd logw), whered is the maximal degree of a vertex,w is the maximal cost of an edge, andn is the size of the tree, andO(n ²), respectively. The problem is shown to be NP-complete for general graphs. Applications of these problems are also discussed.Udi Manber was supported in part by an NSF Presidential Young Investigator Award (Grant DCR-8451397), with matching funds from AT&T.     

Going weighted: Parameterized algorithms for cluster editing

The goal of the Cluster Editing problem is to make the fewest changes to the edge set of an input graph such that the resulting graph is a disjoint union of cliques. This problem is NP-complete but recently, several parameterized algorithms have been proposed. In this paper, we present a number of surprisingly simple search tree algorithms for Weighted Cluster Editing assuming that edge insertion and deletion costs are positive integers. We show that the smallest search tree has size O(1.82^k) for edit cost k, resulting in the currently fastest parameterized algorithm, both for this problem and its unweighted counterpart. We have implemented and compared our algorithms, and achieved promising results.¹     

Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k ^O(dk) n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain K _h as a topological minor, we give an improved algorithm for the problem with running time (O(h)) ^hk n. For graphs which are K _h -minor-free, the running time is further reduced to (O(log h)) ^hk/2 n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(nlog n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs. A preliminary version of this paper appeared in the Proceedings of the 13th Annual International Computing and Combinatorics Conference (COCOON), Banff, Alberta, Canada (2007), pp. 394–405. N. Alon research supported in part by a grant from the Israel Science Foundation, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. This paper forms part of a Ph.D. thesis written by S. Gutner under the supervision of Prof. N. Alon and Prof. Y. Azar in Tel Aviv University.     

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.     

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.     

Maximum weight k-independent set problem on permutation graphs

Based on a Directed Acyclic Graph approach, an O(kn ²) time sequential algorithm is presented to solve the maximum weight k-independent set problem on weighted-permutation graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph. This problem has many applications in practical problems like k-machines job scheduling problem, k-colourable subgraph problem, VLSI design and routing problem.     

Coloring inductive graphs on-line

In this paper we consider the problem of on-line graph coloring. In an instance of on-line graph coloring, the nodes are presented one at a time. As each node is presented, its edges to previously presented nodes are also given. Each node must be assigned a color, different from the colors of its neighbors, before the next node is given. LetA(G) be the number of colors used by algorithmA on a graphG and letx(G) be the chromatic number ofG. The performance ratio of an on-line graph coloring algorithm for a class of graphsC is max_{G C(A(G)/(G))}. We consider the class ofd-inductive graphs. A graphG isd-inductive if the nodes ofG can be numbered so that each node has at mostd edges to higher-numbered nodes. In particular, planar graphs are 5-inductive, and chordal graphs arex(G)-inductive. First Fit is the algorithm that assigns each node the lowest-numbered color possible. We show that ifG isd-inductive, then First Fit usesO(d logn) colors onG. This yields an upper bound ofo(logn) on the performance ratio of First Fit on chordal and planar graphs. First Fit does as well as any on-line algorithm ford-inductive graphs: we show that, for anyd and any on-line graph coloring algorithmA, there is ad-inductive graph that forcesA to use (d logn) colors to colorG. We also examine on-line graph coloring with lookahead. An algorithm is on-line with lookaheadl, if it must color nodei after examining only the firstl+i nodes. We show that, forl/logn, the lower bound ofd logn colors still holds.This research was supported by an IBM Graduate Fellowship.     

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 efficient distributed algorithm for finding all hinge vertices in networks

Let G=(V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v∈V is called a hinge vertex if there exist two vertices in V\{v} such that their distance becomes longer when v is removed. In this paper, we present a distributed algorithm that finds all hinge vertices on an arbitrary graph. The proposed algorithm works for named static asynchronous networks and achieves O(n ²) time complexity and O(m) message complexity. In particular, the total messages exchanged during the algorithm are at most 2m(log n+nlog n+1) bits.     

Complexity and Approximations for Multimessage Multicasting

We consider multimessage multicasting over thenprocessor complete (or fully connected) static network (MM_C). First we present a linear time algorithm that constructs for every degreedproblem instance a communication schedule with total communication time at mostd², wheredis the maximum number of messages that each processor may send or receive. Then we present degreedproblem instances such that all their communication schedules have total communication time at leastd². We observe that our lower bound applies when the fan-out (maximum number of processors receiving any given message) is huge, and thus the number of processors is also huge. Since this environment is not likely to arise in the near future, we turn our attention to the study of important subproblems that are likely to arise in practice. We show that when each message has fan-outk=1 theMM_Cproblem corresponds to the makespan openshop preemptive scheduling problem which can be solved in polynomial time and show that fork?2 our problem is NP-complete and remains NP-complete even when forwarding is allowed. We present an algorithm to generate a communication schedule with total communication time 2d−1 for any degreedproblem instance with fan-outk=2. Our main result is anO(q·d·e) time algorithm, wheree?nd(the input length), with an approximation bound ofqd+k^1/q(d−1), for any integerqsuch thatk>q?2. Our algorithms are centralized and require all the communication information ahead of time. Applications where all of this information is readily available include iterative algorithms for solving linear equations, and most dynamic programming procedures. The Meiko CS-2 machine and computer systems with processors communicating via dynamic permutation networks whose basic switches can act as data replicators (e.g.,nbynBenes network with 2 by 2 switches that can also act as data replicators) will also benefit from our results at the expense of doubling the number of communication phases.