Locally constrained graph homomorphisms—structure,complexity, and applications

A graph homomorphism is an edge preserving vertex mapping between two graphs. Locally constrained homomorphisms are those that behave well on the neighborhoods of vertices. If the neighborhood of any vertex of the source graph is mapped bijectively (injectively, surjectively) to the neighborhood of its image in the target graph, the homomorphism is called locally bijective (injective, surjective, respectively). We show that this view unifies issues studied before from different perspectives and under different names, such as graph covers, distance constrained graph labelings, or role assignments. Our survey provides an overview of applications, complexity results, related problems, and historical notes on locally constrained graph homomorphisms.     

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.     

New Plain-Exponential Time Classes for Graph Homomorphism

A homomorphism from a graph G to a graph H (in this paper, both simple, undirected graphs) is a mapping f:V(G)→V(H) such that if uv∈E(G) then f(u)f(v)∈E(H). The problem Hom (G,H) of deciding whether there is a homomorphism is NP-complete, and in fact the fastest known algorithm for the general case has a running time of O ^*(n(H) ^cn(G)) (the notation O ^*(⋅) signifies that polynomial factors have been ignored) for a constant 0<c<1. In this paper, we consider restrictions on the graphs G and H such that the problem can be solved in plain-exponential time, i.e. in time O ^*(c ^n(G)+n(H)) for some constant c.     

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.     

Complexity of Finding Graph Roots with?Girth Conditions

Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H ², however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G=H ² for some graph H of small girth. The main results are the following.

• There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism.
• There is a polynomial time algorithm to recognize if G=H ² for some graph H of girth at least 6.
• It is NP-complete to recognize if G=H ² for some graph H of girth 4.

These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.     

Edge-Packing in Planar Graphs

Maximum G Edge-Packing (EPack _G ) is the problem of finding the maximum number of edge-disjoint isomorphic copies of a fixed guest graph G in a host graph H . This paper investigates the computational complexity of edge-packing for planar guests and planar hosts. Edge-packing is solvable in polynomial time when both G and H are trees. Edge-packing is solvable in linear time when H is outerplanar and G is either a 3-cycle or a k -star (a graph isomorphic to K _1,k ). Edge-packing is NP-complete when H is planar and G is either a cycle or a tree with edges. A strategy for developing polynomial-time approximation algorithms for planar hosts is exemplified by a linear-time approximation algorithm that finds a k -star edge-packing of size at least half the optimal. Received May 1995, and in revised form November 1995, and in final form December 1997.     

Finding vertex-surjective graph homomorphisms

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes \({{\mathcal G}}\) and \({{\mathcal H}}\), respectively. We determine to what extent a certain choice of \({{\mathcal G}}\) and \({{\mathcal H}}\) influences its computational complexity. We observe that the problem is polynomial-time solvable if \({{\mathcal H}}\) is the class of paths, whereas it is NP-complete if \({{\mathcal G}}\) is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if \({{\mathcal G}}\) is the class of trees and \({{\mathcal H}}\) is the class of trees with at most k leaves, or if \({{\mathcal G}}\) and \({{\mathcal H}}\) are equal to the class of graphs with vertex cover number at most k.     

Backbone coloring of planar graphs without special circles

In this paper, we prove that if G is a connected planar graph that is C₆-free or C₇-free and without adjacent triangles, then there exists a spanning tree T of G such that χ_b(G,T)≤4.     

Additive tree 2-spanners of permutation graphs

Let G be a connected graph. A spanning tree T of G is a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. If their distances in T and G differ by at most t, then T is an additive tree t-spanner of G. In this paper, we show that any permutation graph has an additive tree 2-spanner, and it can be found in O(n) time sequentially or in O(log n) time with O(n/log n) processors on the EREW PRAM computational model by using a previously published algorithm for finding a tree 3-spanner of a permutation graph.     

Labeling Schemes for Tree Representation

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling ℒ _all gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling ℒ _up labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M _up (T) and S _up (T) for ℒ _up and M _all (T) and S _all (T) for ℒ _all . The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0,…,deg (v)−1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for M _up (T), S _up (T) and S _all (T) but NP-hard for M _all (T) (even for 3-regular planar graphs). We show that for every graph G and port labeling there exists a spanning tree T for which S _up (T)=O(nlog log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port labeling. We conclude by discussing some applications for our tree representation schemes. A preliminary version of this paper has appeared in the proceedings of the 7th International Workshop on Distributed Computing (IWDC), Kharagpur, India, December 27–30, 2005, as part of Cohen, R. et al.: Labeling schemes for tree representation. In: Proceedings of 7th International Workshop on Distributed Computing (IWDC), Lecture Notes of Computer Science, vol. 3741, pp. 13–24 (2005). R. Cohen supported by the Pacific Theaters Foundation. P. Fraigniaud and D. Ilcinkas supported by the project “PairAPair” of the ACI Masses de Données, the project “Fragile” of the ACI Sécurité et Informatique, and by the project “Grand Large” of INRIA. A. Korman supported in part by an Aly Kaufman fellowship. D. Peleg supported in part by a grant from the Israel Science Foundation.     

Oriented colorings of 2-outerplanar graphs

A graph G is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the outer face is outerplanar. The oriented chromatic number of an oriented graph H is defined as the minimum order of an oriented graph H^′ such that H has a homomorphism to H^′. In this paper, we prove that 2-outerplanar graphs are 4-degenerate. We also show that oriented 2-outerplanar graphs have a homomorphism to the Paley tournament QR₆₇, which implies that their (strong) oriented chromatic number is at most 67.     

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.     

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.     

Complexity of homomorphisms to direct products of graphs

For a graph G, OALG asks whether or not an input graph H together with a partial map g:S→G, S⊆V(H), admits a homomorphism f:H→G such that f|_S=g. We show that for connected graphs G₁, G₂, OAL G₁×G₂ is in P if G₁ and G₂ are trees and NP-complete otherwise.     

A Complexity Dichotomy For Hypergraph Partition Functions

We consider the complexity of counting homomorphisms from an r-uniform hypergraph G to a symmetric r-ary relation H. We give a dichotomy theorem for r > 2, showing for which H this problem is in FP and for which H it is #P-complete. This generalizes a theorem of Dyer and Greenhill (2000) for the case r = 2, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation H is weighted, and the goal is to compute the partition function, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of r sites interact symmetrically. In the weighted case, our dichotomy theorem generalizes a result of Bulatov and Grohe (2005) for graphs, where r = 2. When r = 2, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all r > 2 the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.     

Subgraphs of 4-Regular Planar Graphs

We shall present an algorithm for determining whether or not a given planar graph H can ever be a subgraph of a 4-regular planar graph. The algorithm has running time O(|H|^2.5) and can be used to find an explicit 4-regular planar graph G⊃H if such a graph exists. It shall not matter whether we specify that H and G must be simple graphs or allow them to be multigraphs.     

Algorithms for Partition of Some Class of Graphs under Compaction and Vertex-Compaction

The compaction problem is to partition the vertices of an input graph G onto the vertices of a fixed target graph H, such that adjacent vertices of G remain adjacent in H, and every vertex and non-loop edge of H is covered by some vertex and edge of G respectively, i.e., the partition is a homomorphism of G onto H (except the loop edges). Various computational complexity results, including both NP-completeness and polynomial time solvability, have been presented earlier for this problem for various classes of target graphs H. In this paper, we pay attention to the input graphs G, and present polynomial time algorithms for the problem for some class of input graphs, keeping the target graph H general as any reflexive or irreflexive graph. Our algorithms also give insight as for which instances of the input graphs, the problem could possibly be NP-complete for certain target graphs. With the help of our results, we are able to further refine the structure of the input graph that would be necessary for the problem to be possibly NP-complete, when the target graph is a cycle. Thus, when the target graph is a cycle, we enhance the class of input graphs for which the problem is polynomial time solvable. We also present analogous results for a variation of the compaction problem, which we call the vertex-compaction problem. Using our results, we also provide important relationships between compaction, retraction, and vertex-compaction to cycles.     

Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

In this paper we provide improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems. Given a graph G=(V,E) with weights w:E→?⁺, a set T ₁,T ₂,…,T _k of subtrees of G is called a tree cover of G if $V=\bigcup_{i=1}^{k} V(T_{i})$ . In the Min-Max k-tree Cover problem we are given graph G and a positive integer k and the goal is to find a tree cover with k trees, such that the weight of the largest tree in the cover is minimized. We present a 3-approximation algorithm for this improving the two different approximation algorithms presented in Arkin et al. (J. Algorithms 59:1–18, 2006) and Even et al. (Oper. Res. Lett. 32(4):309–315, 2004) with ratio 4. The problem is known to have an APX-hardness lower bound of $\frac{3}{2}$ (Xu and Wen in Oper. Res. Lett. 38:169–173, 2010). In the Bounded Tree Cover problem we are given graph G and a bound λ and the goal is to find a tree cover with minimum number of trees such that each tree has weight at most λ. We present a 2.5-approximation algorithm for this, improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).     

Oriented vertex and arc colorings of outerplanar graphs

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping φ from V(G) to V(H), that is φ(x)φ(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. The oriented chromatic index of G is the minimum order of an oriented graph H such that the line-digraph of G has a homomorphism to H.In this paper, we determine for every k?3 the oriented chromatic number and the oriented chromatic index of the class of oriented outerplanar graphs with girth at least k.