Reconstructing Trees from Digitally Convex SetsI

Philip Lafrance, Ortrud R. Oellermann 1, Timothy Pressey

The University of Winnipeg, 515 Portage Avenue, Winnipeg, MB, R3B 2E9

Abstract

Suppose V is a finite set and C a collection of subsets of V that contains ∅and V and is closed under taking intersections. Then C is called a convexityand the ordered pair (V, C) is called an aligned space and the elements of C arereferred to as convex sets. For a set S ⊆ V , the convex hull of S relative to C,denoted by CHC(S), is the smallest convex set containing S. A set S of verticesin a graph G with vertex set V is digitally convex if for every vertex v ∈ V ,N [v] ⊆ N [S] implies v ∈ S. It is shown that every tree is uniquely determinedby its digitally convex sets. These ideas can be used to show that every graphwith girth at least 7 is uniquely determined by its digitally convex sets. Giventhe digitally convex sets of a graph it can be determined efficiently, as a functionof the number of convex sets, if these are those of a tree.

Keywords: Digital convexity, reconstructing graphs from their family ofdigitally convex sets, recognizing families of digitally convex sets of trees

1. Introduction

Reconstructing a graph from partial information about its structure datesback to at least 1941 when P.J. Kelly and S.M. Ulam (see [11] and [19]) con-jectured that every graph of order at least 3 can be determined uniquely, upto isomorphism, from the collection of its 1-vertex deleted subgraphs. Hararyformulated and studied the edge-analogue of this conjecture in [10]. While thereconstruction problem of Kelly, Ulam and Harary deal with partial informa-tion of unlabeled graphs, the reconstruction problem we consider here deals withpartial information of labeled graphs.

Inferring global properties from local information has been the subject ofmuch research, see for example, [15]. With the increased use of the internet andsocial networking sites the problem of determining the behaviour and structure

IDedicated to Andreas Brandstadt on his 65th birthdayEmail addresses: philip_lafrance@hotmail.com (Philip Lafrance),

o.oellermann@uwinnipeg.ca (Ortrud R. Oellermann ), timothy.pressey@gmail.com(Timothy Pressey)

1Research supported by an NSERC Grant CANADA

Preprint submitted to Elsevier May 21, 2014

of these large networks from local information continues to generate interestingquestions and results. For example, the problem of reconstructing a graph fromeither its neighbourhood lists, closed neighbourhood lists, or other neighbour-hood data was considered by Aigner and Triesch in [1], Fomin et al. in [9] andErdos et al. in [7], respectively.

Motivated by determining the structure of chemical compounds from localinformation, Levenshtein [13] posed the problem of finding conditions underwhich a labeled graph is uniquely determined from the radius r metric ballsaround its vertices. In [14] the specific case of radius 2 metric balls is considered.The object of this paper is to study the problem of reconstructing a graph (orimage) from its ‘digitally convex’ sets – these too are defined in terms of localinformation.

We begin by introducing some useful definitions. Suppose V is a finite setand C a collection of subsets of V that contains ∅ and V , and is closed undertaking intersections. Then C is called a convexity or an alignment and theordered pair (V, C) is called an aligned space. The elements of C are referred toas convex sets. For a set S ⊆ V , the convex hull of S, denoted by CHC(S), isthe smallest convex set containing S. An extensive treatment of abstract convexstructures appears in [20].

Euclidean convexity is usually defined in terms of intervals and has a naturalextension to graphs (see [8]): a set S of vertices of a graph is g-convex if forevery pair a, b of vertices of S, every vertex of G that lies on some a–b geodesic(i.e., shortest a–b path) belongs to S. The collection of all vertices that lie onsome a–b geodesic is called the geodesic interval between a and b. Thus a setS of vertices of a graph is g-convex if it contains the geodesic interval betweenevery pair of points in S. The collection of all g-convex sets of a graph forma convexity called the geodesic convexity. Induced paths between a pair a, b ofvertices give rise to another interval notion. The collection of all vertices thatlie on some induced a–b path is called the monophonic interval between a andb. A set S of vertices of a graph is m-convex if it contains the monophonicinterval between every pair of points in S. The collection of all m-convex setsof a graph form a convexity called the monophonic convexity. An extension ofthe geodesic convexity was introduced in [3]. Let G be a connected graph, andlet X be a set of at least two vertices in G. A connected subgraph of smallestsize that contains X is called a Steiner tree for X. The collection of all verticesbelonging to some Steiner tree for X is called the Steiner interval for X. For aninteger k ≥ 2, a set S of vertices in a graph G is said to be k-Steiner convex (orgk-convex ) if the Steiner interval of every subset of k vertices of S is containedin S. Thus S is g2-convex if and only if it is g-convex. Other graph convexities,defined in terms of some type of interval notions have, for example, been studiedin [4], [6], and [16].

The digital convexity of a graph was introduced in [18], mainly as a tool inimage processing to filter digital images. To define this concept we require thefollowing definitions: The neighbourhood N(v) of a vertex v in a graph is thecollection of all vertices adjacent with v and the closed neighbourhood N [v] ofv is N(v) ∪ v. For a set S of vertices in a graph the closed neighbourhood of S

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is the set N [S] = ∪{N [v]|v ∈ S}. A set S ⊆ V is digitally convex if for everyv ∈ V , N [v] ⊆ N [S] implies v ∈ S. The collection of all digitally convex sets ofG, denoted by D(G) or D if G is clear from context, contains ∅ and V and isclosed under intersections. Hence this collection of subsets of V is a convexitywhich we will refer to as the digital convexity. Fig. 1 shows pixels of a digitalimage and the corresponding digital convex hull.

Figure 1: A digital image and its convex hull

Convexity-type parameters for the digital convexity are closely related todomination parameters in a graph. These relationships have been explored forexample in [2] and [17]. If S is a set of vertices in a graph G such that for somev ∈ V (G), N [v] ⊆ N [S] we say S is a local dominating set for v. Thus a set ofvertices is digitally convex if it contains all the vertices for which it is a localdominating set.

Graph convexities usually depend on the structure of a graph and give riseto families of subsets of the vertex set of a graph. It is natural to ask if thegraph can be uniquely determined from the convex sets. Any convexity on thevertex set of a graph uniquely determines its vertex set, since this is precisely thelargest convex set in the family of convex sets. So it is natural to ask if the edgesare also uniquely determined by a given convexity. If a graph can be uniquelydetermined from its convex sets we say that the graph can be reconstructed fromits convex sets. We call this the reconstruction problem for a given convexity.For the geodesic and monophonic convexities of connected graphs this is a trivialtask since the edges are precisely the 2-element convex sets. However, not everyconvexity on the vertex set of a (connected) graph allows one to reconstructthe graph from its convex sets. For example, consider the claw with centrevertex v and leaves x, y and z and the paw with centre vertex v and remainingvertices x, y and z, shown in Fig. 2. For this labeling of the vertices of theclaw and paw, the 3-Steiner convexity for either graph is the family of sets{∅, {v}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {v, x}, {v, y}, {v, z}, {v, x, y}, {v, x,z}, {v, y, z}, {v, x, y, z}}. Indeed this pair belongs to an infinite family of pairsof non-isomorphic graphs that have the same 3-Steiner convexity: To see this,take the claw and the paw and join the same number of leaves to the centrevertex of each to obtain a pair of graphs that has the same 3-Steiner convexity.

For the digital convexity this reconstruction problem is non-trivial and nopairs of non-isomorphic graphs are known that have the same family of digitallyconvex sets. In this paper we solve the problem of uniquely reconstructing atree from its digitally convex sets. It is pointed out that this approach can be

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x y z

v

Claw

zyx

v

Paw

Figure 2: Graphs with the same 3-Steiner convexity

used to reconstruct all graphs of girth at least 7 from their digitally convex sets.Moreover, given the family of digitally convex sets of a graph, we describe aprocedure for determining whether they are those of a tree. We follow [5] forbasic graph theory terminology.

2. Preliminaries

In this section we introduce additional required terminology and some usefulknown results. A vertex is simplicial if its neighbourhood induces a completegraph. A vertex v of a graph G has a private neighbour with respect to a set Sof vertices in G if NG[v] − NG[S − {v}] = ∅. Note that v is not necessarily inS. If v is a leaf in a graph G, then the neighbour of v in G is called the supportvertex of V . Let nD(G) denoted the number of digitally convex sets of a graphG.

The following properties of the digital convexity are established in [12].

Theorem 2.1. [12]

1. If S is a digitally convex set in some graph G, then φ(S) = V (G)−N [S]is digitally convex and φ is a bijection from D(G) to itself.

2. Every graph G has an even number of digitally convex sets.

3. No vertex v ∈ V (G) appears in more than half the digitally convex sets ofG.

4. A vertex v appears in exactly half the digitally convex sets of a graph G ifand only if v is simplicial.

5. A vertex v of a tree T belongs to half the digitally convex sets of T if andonly if v is a leaf. Moreover if v is not a leaf, then v belongs to fewer thanhalf the digitally convex sets of T .

For each integer n ≥ 2 define the spiderstar Sn as follows: (i) If n = 2k + 1is odd, let Sn be the tree obtained from the star K1,k by subdividing each edgeexactly once. (ii) If n = 2k is even, let Sn be the tree obtained from the starK1,k by subdividing all but one of its edges exactly once.

Sharp bounds for the number of digitally convex sets for trees are establishedin [12].

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Theorem 2.2. [12]If T is a tree of order n ≥ 2, then

for even n 2 · 2n/2 − 2

for odd n 3 · 2(n−1)/2 − 2

}≤ nD(T ) ≤ 2n−1.

Moreover the upper bound is attained by the star K1,n−1 and the lower boundsare attained by the spiderstars Sn.

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1 4 5

2 3

Figure 3: The graph S6

The family of digitally convex sets for the spider star of order 6, shown inFig. 3, is D(S6) = {∅, {1}, {2}, {3}, {1, 6}, {2, 4}, {3, 5}, {2, 3}, {1, 2, 4}, {1, 3, 5},{1, 2, 4, 6}, {1, 3, 5, 6}, {2, 3, 4, 5}{1, 2, 3, 4, 5, 6}}. It is easily seen that if T is atree of order at least 3, and v is a leaf of T , then CHD({v}) = {v} and if v isa non-leaf vertex, then CHD({v}) consists of v and its leaf neighbours. So, forexample, in S6, CHD({1}) = {1} and CHD({6}) = {1, 6}.

3. Reconstructing a Tree from its Digitally Convex Sets

In this section we show that trees can be reconstructed from their digitallyconvex sets. We begin by introducing some necessary terminology. If v is avertex in a graph G, then the 2-neighbourhood of v is the set N2[v] = {u ∈V (G)|dG(u, v) ≤ 2}. Let G be a non-trivial connected graph and v a vertex ofG. A set S of vertices, not containing v, is a minimal local dominating set forv if NG[v] ⊆ NG[S] and if for every u ∈ S, NG[v] ⊆ NG[S − {u}].

If v is a vertex of a tree T , the set of neighbours of v in T form a minimallocal dominating set for v. Moreover, if S is a minimal local dominating set forv in T that does not contain v, then S contains a neighbour of v and for eachu ∈ N(v), S contains at least one vertex of N [u]− {v}.

Theorem 3.1. If the digitally convex sets D(T ) of a tree T are given, then theneighbourhood of any vertex v of T can be determined. Thus the digitally convexsets of a tree uniquely determine the tree.

Proof. Let T be a tree and v a vertex of T . We begin by showing how deg(v)can be determined. Vertex v is a leaf that is adjacent to a vertex u if and onlyif v belongs to the convex hull of u.

Suppose now that v is not a leaf. Then deg(v) ≥ 2 and the neighbours of vbelong to distinct components of T −v. We begin by making some observations.

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Suppose S is a minimal local dominating set for v that does not contain v. ThenS must contain all the neighbours of v that are leaves (if any). These can bedetermined by examining CHD({v}), i.e., the smallest convex set that containsv. If this set contains vertices other than v, these are the leaves adjacent withv. If CHD({v}) − {v} is not convex, then T is a star. If CHD({v}) − {v} isconvex, then the leaves adjacent with v are not the only neighbours of v in T .In this case let X be a smallest convex set that contains v but for which X−{v}is not convex. For each vertex w = v in X whose convex hull contains at leasttwo vertices (these are support vertices in X that are not equal to v), replacethe vertices of X that belong to CHD({w}) by w. Let S be the resulting set.Then CHD(S) = X and by our choice of X, S must contain exactly one vertexof N [u]− {v} for all neighbours u of v. So |S − {v}| = deg(v). Let d = deg(v).

Let Sv be the union of all minimal local dominating sets for v. The set Sv

can be found by taking v together with the union of all d-element subsets ofvertices of T −v that are minimal local dominating sets for v. To check if a d-setis a minimal local dominating for v, determine if v ∈ CHD(S) and, if so, addthe elements of S to Sv. We now show how these sets can be used to determinethe neighbours of every vertex in T .

From the above we know how to determine the leaves of T as well as theirneighbours. Assume that v is not a leaf. Consider Sv. We claim that Sv = N2[v].Let u ∈ N2[v] − {v}. Let S = ({u} ∪ N [v]) − N(u) − {v}. Then S is a localdominating set for v, but S −{u} is not a local dominating set for v, as v has aprivate neighbour in the closed neighbourhood of u. Moreover if w ∈ S − {u},then S − {w} is not a local dominating set for v. So S is a minimal localdominating set for v. Thus S ⊆ Sv. Hence N2[v] ⊆ Sv. It is also clear that ifu ∈ V (T ) − N2[v], then for any set S which locally dominates v, S − {u} willalso locally dominate v. So Sv does not contain any vertices that are not inN2[v]. Thus Sv ⊆ N2[v].

We now show how Sv can be used to determine the neighbours of v in T .Construct a set Nv as follows. Begin by adding all the neighbours of v to Nv

that are leaves. For every vertex w of Sv − Nv − {v} if w is not a leaf and ifw is in CHD(Sv − {w}), then add w to Nv. Then Nv = N(v). This followsfrom the fact that a non-leaf neighbour w of v is in the convex hull of Sv −{w}and if w is a non-leaf vertex in N2[v]−N [v], then w is not in the convex hull ofSv − {w}.

4. Determining if the Digitally Convex Sets of a Graph are those ofa Tree

This section is devoted to describing a procedure that determines efficientlywhether a given collection of digitally convex sets of a graph are those of a tree.For a graph G and vertex v of G, let Sv consists of all vertices that belong tosome minimal local dominating set for v.

Lemma 4.1. Let G be a graph and v a vertex of G. Then N [v] ⊆ Sv ⊆ N2[v].

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Proof. If v is an isolated vertex of G, then N [v] = {v} = Sv = N2[v]. Assumenow that v has positive degree. Let u ∈ N(v) and let S = N(v)−N(u). Thenu ∈ S and v ∈ CHD(S). Thus some subset S′ of S will be a minimal localdominating set for v. By the definition of S, u is the only vertex of N [u] that isin N [S]. Since S′ is a minimal local dominating set for v, it follows that u ∈ S′.Thus N [v] ⊆ Sv. No vertex distance at least 3 from v can be in a minimal localdominating set for v. Hence Sv ⊆ N2[v].

Lemma 4.2. A vertex w ∈ N2[v] − N [v] is not in Sv if and only if for everyu ∈ N [v] ∩N [w], there exists some vertex u′ ∈ N [v]−N [w] such that u locallydominates u′, i.e., such that N [u′] ⊆ N [u].

Proof. We begin by proving the sufficiency of the statement by the contrapos-itive. Suppose there is some vertex u in N [w] ∩ N [v] which does not locallydominate any vertex in N [v] −N [w]. So every vertex of N [u] ∩ (N [v] −N [w])has a private neighbour with respect to N [u]. Let S = (N2[v] − N [u]) ∪ {w}.Thus N [u]∩S = {w}. Thus no vertex of S−{w} locally dominates u. We nowshow that N [S] contains N [v]. Let x ∈ N [v]. If x ∈ N [S] − N [u], then x ∈ Sand thus x ∈ N [S]. If x ∈ N [u], then, by assumption, x has a private neighbourwith respect to N [u]. So x ∈ N [S]. So S is a local dominating set for v. HenceS contains a minimal local dominating set for v. Such a set necessarily containsw, since w is the only vertex of S adjacent with u. So w ∈ Sv. This completesthe proof of the sufficiency.

For the converse suppose that w ∈ N2[v] − N [v] is in some minimal localdominating set S for v. So N [v] ⊆ N [S] but no proper subset of S has thisproperty. So N [v] ⊆ N [S − {w}]. Hence there is some u ∈ N [v] such thatN [u] ∩ S = {w}. Thus u = v. We show next that there is no u′ ∈ N [v]−N [w]such that N [u′] ⊆ N [u]. Assume, to the contrary, that there is some u′ ∈N [v] − N [w] such that N [u′] ⊆ N [u]. Since no vertex of S − {w} is adjacentwith u, N [u′] ∩ S = ∅. So N [u′] ⊆ N [S], which is not possible since u′ ∈ N [v]and N [v] ⊆ N [S]. This completes the proof of the necessity.

Lemma 4.3. Let v be a vertex of a graph G. Then Sv = N2[v] if and only iffor every w ∈ N2[v] − N [v], and every u ∈ N [w] ∩ N [v], the vertex u does notlocally dominate any vertex of N [v]−N [w].

Proof. This follows immediately from Lemmas 4.1 and 4.2.

We consider next the problem of determining whether a given collection ofdigitally convex sets of a graph is the collection of digitally convex sets of a tree.

Lemma 4.4. Let G be a graph. Suppose G has the following property P: Avertex w is a leaf whenever w is locally dominated by some vertex u. ThenSv = N2[v] for every non-leaf vertex v.

Proof. Suppose that Sv = N2[v] for some non-leaf vertex v. Let w ∈ N2[v]−Sv.By Lemma 4.2, there is for every u ∈ N [v] ∩N [w] some u′ ∈ N [v]−N [w] suchthat u′ is locally dominated by u. Since u′ is adjacent with both u and v, it

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follows that u′ is not a leaf. So G does not have property P. The result nowfollows.

Suppose D(G) is the collection of digitally convex sets of some graph G withvertex set V . We construct a digraph DG, called the local dominating digraphfor G, as follows. The vertex set of DG is the same as the vertex set of G; anda vertex u is adjacent to a vertex v in DG if and only if u = v and v belongsto the convex hull of u, i.e. if N [v] ⊆ N [u]. The in-degree of a vertex v in DG,denoted by idDG

(v), equals the number of vertices that locally dominate v.

Lemma 4.5. If T is a tree of order at least 2, then every non-leaf vertex of Thas in-degree 0 in DT and a leaf of T has in-degree 1 in DT .

Proof. Since a leaf of a tree is locally dominated by precisely its support vertex,a leaf has in-degree 1 in DT . If v is a vertex of a tree that is not a leaf, thendeg(v) ≥ 2. Since a tree has no cycles, no neighbour of v is adjacent with anyother neighbour of v. Hence no vertex of T − v locally dominates v.

Proposition 4.6. A vertex v of a graph G is a simplicial vertex of degree kif and only if v belongs to half of the digitally convex sets of G and if v hasin-degree k in DG.

Proof. Suppose v is a simplical vertex of degree k in G. By Theorem 2.1, vbelongs to half the number of digitally convex sets of G. Moreover, since N [v] iscontained in the closed neighbourhood of each of its neighbours, the in-degreeof v in DG is at least k. Moreover, if N [v] ⊆ N [u], then u is a neighbour of v.Hence idDG

(v) = k.Conversely suppose that v belongs to half the convex sets of G and that

idDG(v) = k. By Theorem 2.1, v is simplicial. Since N [v] ⊆ N [u] for every

neighbour u of v, idDG(v) ≥ degG(v). Clearly there are at most degG(v) neigh-

bours of v whose closed neighbourhood contains N [v]. So idDG(v) ≤ degG(v).

The result now follows.

Corollary 4.7. A vertex v of a graph G is a leaf if and only if v lies in half ofthe digitally convex sets of G and if v has in-degree 1 in DG.

Corollary 4.8. If the maximum in-degree of DG is 1 and if every vertex of DG

with in-degree 1 is a leaf, then N2[v] = Sv for every non-leaf vertex v of G.

Proof. This follows from Lemma 4.4 and Corollary 4.7.

We say that a proper subgraph H of a graph G is an isolated subgraph of Gif no vertex of H is adjacent with any vertex of G−V (H). So H is a componentor a union of components of G.

Lemma 4.9. A subgraph H of G is an isolated subgraph of G if and only if forevery convex set S in G− V (H), the set S ∪ V (H) is also convex.

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Proof. The sufficiency of the condition is easily seen to hold. For the conversesuppose that H is not a component of G. Then some vertex v of H is adjacentto some vertex u of V (G)− V (H). Observe that CHD(V (H)) ⊆ N [V (H)] andthat N [CHD(V (H))] = N [V (H)]. From an earlier result φ(V (H)) = V (G) −N [V (H)] is convex. Thus by the hypothesis φ(V (H)) ∪ V (H) is convex inG. However u ∈ φ(V (H)) ∪ V (H) and N [u] ⊆ N [φ(V (H)) ∪ V (H)]. Thiscontradicts the fact that φ(V (H)) ∪ V (H) is convex. Thus H is an isolatedsubgraph of G.

Let (V,D) be an aligned space. A convex set X is a half-space if V −X isalso convex.

Corollary 4.10. If H is a component of G that is a proper subgraph of G, thenH is an isolated subgraph of G and V (H) is a half-space.

Algorithm 4.11. Given the collection D(G) of digitally convex sets of a graphG = (V,E), this algorithm efficiently determines if G is a tree.

1. Let d = |D(G)| and let D ← D(G).

2. If n is even and d < 2 ·2n/2−2, output G is not a tree and stop. Otherwiseproceed to Step 4.

3. If n is odd and d < 3 · 2(n−1)/2 − 2, output G is not a tree and stop.Otherwise proceed to Step 4.

4. Delete ∅ and V from D.5. If D = ∅ proceed to Step 7 as G is connected. Otherwise let X ∈ D. If

V −X is not digitally convex, delete X from D and repeat Step 5; otherwisecontinue to Step 6.

6. If for each Y ∈ D(G), Y ∩X = ∅ implies Y ∪X ∈ D(G), then output Gis not a tree since G is not connected and stop. Otherwise delete X andV −X from D and go to Step 5.

7. Construct the digraph DG. If the in-degree of any vertex of DG exceeds 1,output G is not a tree and stop. Otherwise, continue to Step 8.

8. If some vertex of in-degree 1 in DG does not lie in half of the convex sets,then output G is not a tree and stop. Otherwise, continue to Step 9.

9. For each non-leaf vertex v of G (i.e. vertex of degree 0 in DG) determineSv - the union of all minimal local dominating sets for v. Let Nv consistof all vertices of CHD({v}) − {v} together with all non-leaf vertices x ∈Sv − {v} such that x ∈ CHD(Sv − {x}). Let dv = |Nv|. For each leafvertex let dv = 1.

10. If∑

v∈V (G) dv = 2n− 2, output G is a tree and stop; otherwise output Gis not a tree and stop.

Theorem 4.12. If D(G) is the collection of digitally convex sets of a graphG = (V,E) of order n, then Algorithm 4.11 determines in polynomial timewhether D(G) is the collection of digitally convex sets of a tree.

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Proof. If the algorithm halts after Steps 2 or 3 are completed, it correctly de-termines that G is not a tree, by Theorem 2.2. If the algorithm halts during anexecution of Step 6, then, by Lemma 4.9 and Corollary 4.10, G is disconnected.So the algorithm correctly indicates that G is not a tree. If the algorithm haltsafter Step 7, it follows from Lemma 4.5 that G is not a tree. If the algorithmhalts after Step 8 is completed, then, by Corollary 4.7, G contains a non-leafvertex that is locally dominated by some other vertex. This is not possible in atree and hence the algorithm correctly determines that G is not a tree.

By Steps 7 and 8 and Corollaries 4.7 and 4.8 it follows that the set Sv

determined in Step 9 is N2[v]. Also if x is a non-leaf neighbour of v, thenx ∈ CHD(Sv − {x}). So x ∈ Nv and thus NG(v) ⊆ Nv. Therefore |Nv| =dv ≥ degG(v). If G is a tree, then Nv = NG(v) as was observed in the proof ofTheorem 3.1.

If the algorithm halts in Step 10 after determining that∑

v∈V (G) dv = 2n−2,then by Step 6 and Lemma 4.9 G is connected and

∑v∈V (G) dv = 2n − 2.

Since∑

v∈V (G) dv ≥∑

v∈V (G) degG(v) ≥ 2n − 2, it follows that∑

v∈V (G) dv =∑v∈V (G) degG(v) = 2n − 2 and that dv = degG(v) for all vertices v of G. So

G is connected and has n− 1 edges. Thus the algorithm correctly outputs thatG is a tree. If

∑v∈V (G) dv = 2n − 2, then either NG(v) = Nv for some vertex

v of G or G has more than n − 1 edges. In either case G is not a tree. So thealgorithm correctly outputs, in this case, that G is not a tree.

We now discuss ways in which each of the steps in the algorithm can beimplemented as well as their resulting complexities. Steps 1-4 can be performedusing O(n) operations.

For the remainder of the algorithm, begin by randomly ordering the verticesof G as v1, v2, . . . , vn. Let X ∈ D(G) and suppose that |X| = k. Order thevertices of X as vi1 , vi2 , . . . , vik where i1 < i2 < . . . < ik. Assigning suchan ordering to each X ∈ D(G) takes O(d n log n) comparisons if a merge sortalgorithm is used. We use this ordering to construct the Hasse digraphH(D(G))for the family D(G) of digitally convex sets of G. The vertices of this digraph arethe elements of D(G) and (X,Y ) is an arc of this digraph if and only if X ⊆ Y .At most O(n

(d2

)), i.e. O(nd2) comparisons are made to construct H(D(G)).

For a given X ∈ D(G) it can be determined with at most dn comparisonswhether V −X ∈ D(G). So the overall complexity of Step 5 is O(nd2).

Step 6 is performed if X and V −X are in D(G). Take every in-neighbourY of VX and check if X ∪Y ∈ D(G). This can be done by checking if a commonout-neighbour of X and Y in H(D(G)) has |X| + |Y | elements. This checkcan be performed with O(nd) comparisons. Overall the complexity of Step 6 isO(nd2).

To implement Step 7 begin by ordering D(G) according to non-decreasing setsize. This can be accomplished by first determining the size of every elementof D(G) in O(nd) steps and then using a merge sort to order the sets usingO(d log d) comparisons. For each vertex v of G determine its convex hull byfinding the smallest set that contains it. This can be done by making O(n2d)comparisons. Now join each vertex v by an arc to each vertex that is in its

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convex hull (other than itself). The resulting graph is DG. The in-degree ofevery vertex in D(G) can be found in O(n2) steps. Overall the complexity ofStep 7 is O(n2d).

Since each convex set has at most n elements, at most nd comparisons aremade to determine whether a vertex of in-degree 1 in DG lies in half the numberof convex sets. So overall the complexity of Step 8 is O(n2d).

To find Sv for a non-leaf vertex v in Step 9 begin by determining all possiblesubset of V − CHD({v}). By Theorem 2.2 there are O(d2) possible subsets.Order these sets from smallest to largest. The complexity of finding this orderingis O(nd2). Let Ov be the resulting ordering. Determine the first set S in thisordering for which the convex hull of S∪(CHD({v})−{v}) contains v. This canbe achieved by using the ordering of the convex sets D(G) found in Step 7 andthe ordering of the vertices of G mentioned above to determine the convex hullof S ∪ (CHD({v})− {v}). Let |S| = k. Add S to the collection Mv of minimaldominating sets for v and determine any other minimal dominating sets for vof cardinality k and add these to Mv. Now let S′ ∈ Ov be such that |S′| > k.If the convex hull of S′ ∪ (CHD({v})− {v}) contains v, determine if no propersubset of S′ is in Mv. If so, then add S′ to Mv; otherwise S′ is not added toMv. The set Sv is the union of all elements in Mv. The overall complexity ofStep 9 is O(nd4).

Step 10 has complexity O(n).Hence the algorithm is polynomial in the number of convex sets and the

order of the graph.

5. Concluding Remarks

The approach used in Theorem 3.1 can be used to reconstruct all graphs ofgirth at least 7 from their digitally convex sets. These graphs can be thought ofas being ‘locally’ tree-like. The proof of Theorem 3.1 hinges on the fact that forfor a vertex v of a tree the set Sv, as defined in this paper, equals N2[v]. It isreadily seen that for graphs of girth at least 7 it is also the case that Sv = N2[v].It was also shown in [14] that all graphs of girth at least 7 can be reconstructedfrom their closed 2 balls, i.e. form N2[v]. However, the problem of determiningthe girth of a graph from its digitally convex sets, is still an open problem as isthe problem of determining whether graphs with cycles and girth at most 6 canbe reconstructed from their digitally convex sets.

Let F be a family of subsets of a finite ground set V and let Π be the problemof deciding whether F is the family of digitally convex sets of a graph or not.Determining whether Π is an NP-complete problem remains open. In order forF to be the digital convexity of a graph, it must contain ∅ and V and be closedunder intersections. Moreover, conditions 2. and 3. of Theorem 2.1 need tohold. By 1. of Theorem 2.1 F can be partitioned into disjoint (ordered) pairs(S, φ(S)) such that if A,B ∈ F with A ⊆ B, then φ(B) ⊆ φ(A). It is not knownwhether these conditions are sufficient.

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Acknowledgements We wish to thank the referees for their useful commentsthat improved the presentation of this paper.

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