Information Processing Letters 114 (2014) 387–391

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Information Processing Letters

www.elsevier.com/locate/ipl

Exact algorithm for graph homomorphism and locallyinjective graph homomorphism

Paweł Rzazewski

Warsaw University of Technology, Faculty of Mathematics and Information Science, Koszykowa 75, 00-662 Warszawa, Poland

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 October 2013Received in revised form 24 February 2014Accepted 24 February 2014Available online 28 February 2014Communicated by Ł. Kowalik

Keywords:Graph algorithmsGraph homomorphismLocally injective homomorphismH(2,1)-labelingExact algorithm

For graphs G and H , a homomorphism from G to H is a function ϕ : V (G) → V (H), whichmaps vertices adjacent in G to adjacent vertices of H . A homomorphism is locally injectiveif no two vertices with a common neighbor are mapped to a single vertex in H . Many casesof graph homomorphism and locally injective graph homomorphism are NP-complete, sothere is little hope to design polynomial-time algorithms for them. In this paper we presentan algorithm for graph homomorphism and locally injective homomorphism working intime O∗((b + 2)|V (G)|), where b is the bandwidth of the complement of H .

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Graphs homomorphism problem (or H-coloring, as itis sometimes called) is a natural generalization of a well-known graph coloring problem. For graphs G and H we saythat ϕ : V (G) → V (H) is a homomorphism from G to H ifϕ(v)ϕ(u) ∈ E(H) for any uv ∈ V (G). In other words, a ho-momorphism is an edge-preserving mapping from V (G)

to V (H). Thus the k-coloring problem is equivalent tothe problem of finding a homomorphism to the completegraph Kk . We refer the reader to the monography by Helland Nešetril [13] for more information about graph homo-morphisms.

Some special cases of graph homomorphism, namely lo-cally constrained graph homomorphisms have also received aconsiderable attention (see the survey by Fiala and Kra-tochvíl [6] for more information about the topic). We saythat the homomorphism ϕ is locally injective (locally sur-jective; locally bijective) if the neighborhood of v ∈ V (G) ismapped injectively (resp.: surjectively; bijectively) to theneighborhood of ϕ(v). We will be mostly interested in

E-mail address: p.rzazewski@mini.pw.edu.pl.

http://dx.doi.org/10.1016/j.ipl.2014.02.0120020-0190/© 2014 Elsevier B.V. All rights reserved.

locally injective homomorphisms. They can be seen as ho-momorphisms from G to H , in which no two vertices fromG with a common neighbor are mapped to the same ver-tex of H .

Homomorphisms and locally constrained homomor-phisms generalize and unify many known problems ingraph theory, e.g. colorings, graph covers, role assign-ments, etc. For example (m,k)-coloring (see Zhu [18]),i.e. the problem of finding an assignment f : V (G) →{0,1, . . . ,m−1}, such that k � | f (v)− f (u)| � m−k when-ever vu ∈ E(G), is equivalent to finding a homomorphismto Ck−1

m , a complement of a (k−1)th power of an m-cycle.1

By a complement of a graph G = (V , E), we mean thegraph G = (V ,

(V2

) \ E).Another example is the so-called H(2,1)-labeling prob-

lem (where H is a graph). In this problem we havetwo constraints: (i) the vertices, which are adjacent in Gare mapped onto distinct, nonadjacent vertices of H ;and (ii) vertices, which have a common neighbor inG , are mapped onto distinct vertices of H . Fiala andKratochvíl [5] showed a close relation between locally

1 Here we consider C0m to be a graph with m vertices and no edges.

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injective homomorphisms and H(2,1)-labelings: anH(2,1)-labeling of G is exactly a locally injective homo-morphism from G to H .

The well-known L(2,1)-labeling problem (see Griggsand Yeh [11]) can be seen as a problem of finding the min-imum k such that the input graph admits an H(2,1)-label-ing for H being a path with k + 1 vertices. Another inter-esting case of H(2,1)-labeling is a circular L(2,1)-labeling,sometimes denoted by Lc(2,1)-labeling (see Liu and Zhu[3]). It is equivalent to finding the minimum k, for whichthe input graph G has an H(2,1)-labeling for H being acycle with k + 1 vertices.

Graph homomorphisms are also interesting from thecomputational point of view. In their celebrated theorem,Hell and Nešetril [14] showed that determining if G hasa homomorphism to H is polynomial if H is bipartite andNP-complete otherwise. For a locally surjective homomor-phism Fiala and Paulusma [9] showed that determiningthe existence of a locally surjective homomorphism fromG to a connected graph H is polynomial if H has at most2 vertices and NP-complete otherwise. They also showeda full dichotomy for the case when H is disconnected, butthe description of polynomial cases is more complicated.There is no similar characterization for the case of locallyinjective homomorphisms and locally bijective homomor-phisms, but still we can find some partial results (see forexample [5,7,8] for some results on locally injective ho-momorphisms and [1,17] for locally bijective homomor-phisms).

As many cases of graph homomorphism and locallyconstrained graph homomorphism are NP-complete, thereis little hope to obtain polynomial algorithms for them.Therefore a natural approach is to design exponentialalgorithms with the basis of the exponential factor ina complexity bound expressed as a function of some in-variant of H . Fomin et al. [10] presented the algorithmfor graph homomorphism from G to H working in timeO∗((2 tw(H) + 1)n),2 where tw(H) denotes a treewidthof the graph H (see Diestel’s book [4] for some infor-mation about treewidth of graphs) and n is the numberof vertices of G . For a locally injective homomorphisms,Havet et al. [12] presented an algorithm working in timeO∗((�(H) − 1)n). To the best of our knowledge there areno similar results for a locally surjective and a locally bi-jective graph homomorphism problem.

In this paper we show how to adapt the algorithm forL(2,1)-labeling by Junosza-Szaniawski et al. [15] to solvethe graph homomorphism and locally injective graph ho-momorphism problems. We have already mentioned thatgraph homomorphism to a complete graph is equivalent tothe graph coloring problem and therefore can be solved intime O∗(2n), using the algorithm by Björklund et al. [2].Finding a locally injective homomorphism to a completegraph can also be reduced to the classical graph coloringproblem, since it is equivalent to coloring a square of thegraph. Therefore in this paper we shall focus on the casewhen H is not a complete graph. The main result of thispaper is the following theorem.

2 In the O∗ notation we suppress polynomial factors.

Theorem 1. The existence of a homomorphism from G to H canbe decided in time O∗((bw(H) + 2)n), where n is the numberof vertices of G and bw(H) is the bandwidth of the complementof H.

After a small adaptation in the algorithm for the graphhomomorphism problem, we obtain the algorithm for thelocally injective homomorphism problem, working withinthe same time bound.

Theorem 2. The existence of a locally injective homomorphismfrom G to H can be decided in time O∗((bw(H)+2)n), where nis the number of vertices of G and bw(H) is the bandwidth of thecomplement of H.

2. Preliminaries

In this paper we consider simple graphs without loopsand multiple edges. If ϕ is a homomorphism from G to H ,we will write ϕ : G → H shortly. A homomorphism is par-tial if we allow that some vertices of G are not mapped toany vertices of H .

Let G be a graph and let L = v1 v2 . . . vn be some order-ing of its vertices. The bandwidth of a graph G , denoted bybw(G), is the minimum over all orderings L of the valuemax{|i − j|: vi v j ∈ E(G)}. Informally speaking, we want toplace the vertices of G on integer points of a number linein such a way, that the longest edge is as short as possible.

For � ∈ N let [�] denote the set {0,1,2, . . . , �}. More-over, let [[�]] denote the set [�] ∪ {0}, where 0 is a specialsymbol, whose meaning will be clarified later.

For a set of vectors A ⊆ Σn and a symbol x ∈ Σ

(for some set Σ of symbols) let Ax denote the set {v ∈Σn−1: xv ∈ A}, where xv denotes a vector obtained by ap-pending x to the front of v (so xv ∈ Σn).

3. Exact algorithm for graph homomorphism

Let us consider a problem of deciding if a graph G hasa homomorphism to H . Let V (G) = {v1, v2, . . . , vn} andV (H) = {h1,h2, . . . ,hm}. The ordering of vertices in G isarbitrary. The vertices of H are arranged in the order cor-responding to the bandwidth of H , i.e. in such a way thatthe value max({|i − j|: hih j /∈ E(H)} is minimum possible(by the definition, this minimum value is equal to bw(H)).Let β = bw(H) + 1 and let Hk = H[{h1,h2, . . . ,hk}] for anyk ∈ {1,2, . . . ,m}.

In this section we prove Theorem 1 by presenting analgorithm for determining the existence of the homomor-phism G → H , working in time O∗((bw(H) + 2)n). Weshall proceed in a way similar to the algorithm by Junosza-Szaniawski et al. [15]. We will use dynamic programmingand try to extend partial homomorphisms from G to Hk topartial homomorphisms from G to Hk+1.

The partial homomorphisms G → Hk are stored ina compact way, unifying those which can be extendedin the same way. Then for every partial homomorphismϕ: G → Hk we shall look for a set X ⊆ V (G), such that:(i) X is independent, (ii) no vertex from X is mapped byϕ , (iii) no neighbor of a vertex from X is mapped to a non-neighbor of hk+1. If such a set exists, we can extend ϕ by

P. Rzazewski / Information Processing Letters 114 (2014) 387–391 389

mapping all vertices from X to hk+1. This way we obtaina partial homomorphism G → Hk+1.

Let P be a set of characteristic vectors of independentsets in G . In other words, p ∈ P if and only if there existsan independent set X in G such that pi = 1 iff vi ∈ X andpi = 0 otherwise.

Now let us describe our compact representation of par-tial homomorphisms from G to Hk . For every k = 1, . . . ,mwe introduce a set T [k] ⊆ [β]n such that a ∈ T [k] if andonly if there exists a partial homomorphism ϕ : G → Hk ,satisfying the following condition:

ai ={

0 if vi is not mapped,1 if ϕ(vi) = h� with 1 � �� k − β + 1,x ∈ {2, . . . , β} if ϕ(vi) = h� with � = k − β + x.

Moreover, let us define T [0] := {0n}. Note that vectorsa with no 0’s correspond to homomorphisms G → Hk .Therefore there exists a homomorphism G → H if and onlyif T [m] ∩ {1,2, . . . , β}n �= ∅.

Now we want to compute sets T [k] (for k ∈ {1, . . . ,m}),using sets T [k − 1] and P . To do this, we shall introducetwo operations. Let k ∈ {1, . . . ,m − 1} be fixed and assumewe have computed the set T [k]. Let a be a vector fromT [k]. Let a ∈ [[β]]n be defined as follows:

ai =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if ai = 0 and there is no v j ∈ NG (vi)

with a j � 2 and hk−β+a j /∈ NH (hk+1),

0 if ai = 0 and there exists v j ∈ NG (vi)

with a j � 2 and hk−β+a j /∈ NH (hk+1),

x ∈ {1, . . . , β} if ai = x.

Consider a ∈ T [k] (for k � m − 1) and a partial homo-morphism ϕ corresponding to a. Notice that if vi is notmapped by ϕ , then ai is either equal to 0 or to 0. Wehave ai = 0 if and only if ϕ can be extended by map-ping vi to hk+1. In other case, ai is equal to 0. Since thevertices of H are arranged according to bw(H), all non-neighbors of hk+1 are in {hk−bw(H)+1,hk−bw(H)+2, . . . ,hk}.

This justifies the unification of the sets of vertices mappedto h1,h2, . . . ,hk−bw(H) in our representation of partial ho-

momorphisms. Let T [k] denote the set {a: a ∈ T [k]}. Notethat computing T [k] from T [k] takes time O(|T [k]| · n2),since in each vector a ∈ T [k] we just have to check all pairsof vertices vi, v j such that vi v j ∈ E(G) and ai = 0. More-over observe that |T [k]| = |T [k]| for every k.

Now let us define the partial function ⊕ : [[β]]×{0,1} →[β] as follows:

x ⊕ y =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if x ∈ {0, 0} and y = 0,

1 if x ∈ {1,2} and y = 0,

x − 1 if x ∈ {3,4, . . . , β} and y = 0,

β if x = 0 and y = 1,

undefined otherwise.

We generalize this operation to vectors coordinate-wise(x1x2 . . . xn ⊕ y1 y2 . . . yn is (x1 ⊕ y1) . . . (xn ⊕ yn) if xi ⊕ yiis defined for all i ∈ {1, . . . ,m} or is undefined other-wise); and sets of vectors: A ⊕ B = {a ⊕ b: a ∈ A, b ∈ B,

a ⊕ b is defined}.Observe that for a ∈ T [k] and p ∈ P , computing a ⊕ p

corresponds to extending a partial homomorphism bymapping all the vertices from the set encoded by p tovertex hk+1. Now we are ready to present the algorithm.

Algorithm 1: Determine-Hom(G, H).

1 P ← a set of characteristic vectors of independent sets of G2 T [0] ← {0n}3 for k ← 1 to m do4 compute T [k − 1] from T [k − 1]5 T [k] ← T [k − 1] ⊕ P

6 if T [m] ∩ {1,2, . . . , β}n �= ∅ then return Yes

7 else return No

To prove the correctness, it is enough to show thatT [k] = T [k − 1] ⊕ P for any k = 1,2, . . . ,m. Let a ∈ T [k]and ϕ : G → Hk be a partial homomorphism correspond-ing to a. Let p be a characteristic vector of the set X :=ϕ−1(hk). This set is clearly independent, so p ∈ P . Letϕ′ : G → Hk−1 be a partial homomorphism obtained byrestricting ϕ to Hk−1 (i.e. the vertices from X are notmapped by ϕ′), and let a′ be a vector in T [k − 1] corre-sponding to ϕ′ . Let vi be a vertex from X . Every neighborof vi has to be mapped to some neighbor of hk or not bemapped at all, so a′

i = 0. We observe that a = a′ ⊕ p andtherefore a ∈ T [k − 1] ⊕ P .

On the other hand, let a′ ∈ T [k − 1] (with a correspond-ing partial homomorphism ϕ′ : G → Hk−1) and p ∈ P . Letus extend ϕ′ to partial homomorphism ϕ : G → Hk withevery vertex vi from the independent set corresponding top mapped to vertex hk . This is possible if and only if eachneighbor of vi is not mapped or is mapped to a neigh-bor of hk . In other words, we require that ai = 0. Thereforea′ ⊕ p ∈ T [k] with corresponding partial homomorphism ϕ .

We shall perform the computation of T [k − 1] ⊕ P re-cursively. We consider vectors starting with each elementfrom [β] separately.

T [k] ⊕ P

=⋃

b∈[[β]],p∈{0,1}s.t. b⊕p is defined

(b ⊕ p)(T [k]b ⊕ P p

)

= 0[(

T [k]0 ∪ T [k]0

) ⊕ P0] ∪ 1

[(T [k]1 ∪ T [k]2

) ⊕ P0]

∪⋃

a∈{2,...,β−1}a[T [k]a+1 ⊕ P0

] ∪ β[T [k]0 ⊕ P1

].

To compute ⊕ on two sets of vectors of length n, wehave to compute ⊕ on β + 1 pairs of sets of vectors oflength n − 1. The size of P is at most n · 2n bits, the sizeof T [k − 1] is at most n · (β + 1)n bits. Recall that comput-ing T [k − 1] takes time at most O(|T [k − 1]| · n2) = O(n3 ·(β + 1)n). Therefore time complexity is given by: F (n) =O(n ·2n +n3 ·(β +1)n +(β +1) · F (n−1)). One can verify byinduction that this recursion is satisfied by F (n) = O(n3 ·2n + n3 · (β + 1)n). Since β = bw(H) + 1, we have β + 1 =bw(H)+2 � 2. Finally we obtain F (n) =O∗((bw(H)+2)n),which proves Theorem 1.

We have mentioned that (m,k)-coloring is equivalent toa homomorphism to Ck−1

m . Since the complement of Ck−1m

is Ck−1m and bw(Ck−1

m ) = 2(k − 1), we obtain the following.

Corollary 1. The (m,k)-coloring problem on a graph G with nvertices can be solved in time O∗((2k)n).

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4. Locally injective homomorphism and H(2,1)-labeling

In this section we prove Theorem 2 by modifying Al-gorithm 1 to determine the existence of a locally injectivehomomorphism from G to H .

We observe that the vertices of G that can be mappedto a single vertex of H (in a locally injective manner) mustbe in a distance at least 3 from each other. Therefore theyform a 2-independent set, which is exactly a set in whichno two vertices are adjacent or have a common neighbor(in other words, it is an independent set in a square of thegraph, see for example [16] for more information aboutsuch sets). Since it is the only additional requirement forlocally injective homomorphisms, the only thing that hasto be changed in Algorithm 1 is the initialization of theset P . Now it has to contain characteristic vectors of all2-independent sets. This proves Theorem 2.

Recall that H(2,1)-labelings are exactly locally injectivehomomorphisms to H . Since the complement of H is H ,we obtain the following corollary.

Corollary 2. For any graphs G and H we can solve theH(2,1)-labeling problem in time O∗((bw(H) + 2)n), wheren is the number of vertices of G.

Let us see how this bound works for the Lc(2,1)-label-ing problem. It is equivalent to finding the smallest m, suchthat the input graph admits a Cm(2,1)-labeling. We shallcheck the existence of such a labeling for m = 3, . . . ,2nand stop when we find one. Since bw(Cm) = 2, we obtainthe following.

Corollary 3. The Lc(2,1)-labeling problem on a graph G with nvertices can be solved in time O∗(4n).

The bounds presented here can be slightly improved,using the methods presented in [16] and [15]. However, itrequires many technical calculations and the improvementgets smaller as bw(H) grows.

5. Concluding remarks and open problems

Let us conclude this paper with some comparison ofour algorithm with the previously known algorithms forthe graph homomorphism problem [10] and the locallyinjective graph homomorphism problem [12]. Recall thattheir complexities are bounded by O∗((2 tw(H) + 1)n) andO∗((�(H) − 1)n), respectively. Observe that both param-eters in these bounds, i.e. the treewidth and the maxi-mum degree, grow rapidly as the density of H increases.Therefore if H is extremely dense (for example obtainedfrom a complete graph by removing a constant numberof edges or a matching), both those algorithms may workvery slowly. But in such cases the graph H is very sparseand its bandwidth is small, so our algorithm outperformsthe previous ones significantly. On the other hand, if H issparse (and therefore H is dense), bw H is large and so isthe complexity bound for our algorithm.

There are still many open questions concerning exactalgorithms for the (locally constrained) graph homomor-phism problems. For example, as we mentioned before,

there are no exact algorithms deciding the existence ofa locally surjective or a locally bijective graph homomor-phism.

Open problem 1. Design a non-trivial exact exponential al-gorithm for the locally surjective and the locally bijectivegraph homomorphism problem.

It still remains a great challenge to design an exactalgorithm deciding the existence of a (locally injective) ho-momorphism from G to H , whose time complexity doesnot depend on H (or show that there is no such algorithm,under some standard complexity assumptions). We conjec-ture that, under the Exponential Time Hypothesis, such analgorithm does not exist.

Open problem 2. Assuming the Exponential Time Hypoth-esis, show that there is no exact algorithm for the (locallyinjective) graph homomorphism problem with time com-plexity bounded by O∗(cn) for a constant c (or design suchan algorithm).

Acknowledgements

The author is sincerely grateful to Konstanty Junosza-Szaniawski for valuable discussion and advice. We alsothank the reviewers for recommending various improve-ments in exposition.

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