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On computational complexity of length embeddability of graphs
Discrete Mathematics 339 (2016) 2605–2612
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
On computational complexity of length embeddability of graphs Mikhail Tikhomirov Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation
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Article history: Received 21 October 2014 Received in revised form 11 May 2016 Accepted 11 May 2016
abstract A graph G is embeddable in Rd if the vertices of G can be assigned to points of Rd in such a way that all pairs of adjacent vertices are at distance 1. We show that verifying embeddability of a given graph in Rd is NP-hard in the case d > 2 for all reasonable notions of embeddability. © 2016 Elsevier B.V. All rights reserved.
Keywords: Computational complexity Graph embedding Distance graph
1. Introduction The unit-distance graph of S ⊂ Rd is defined as the graph G = (V , E ), where V = S and E is the set of all pairs of points x, y ∈ S such that x and y are at distance 1. A graph is a unit-distance graph in Rd if it is isomorphic to the unit-distance graph of some set S ⊂ Rd . Some famous problems concerning unit-distance graphs are the Erdős’ unit distance problem on the maximal number of unit distances between n points in R2 (see [1,4,2]), the Hadwiger–Nelson problem on the chromatic number of R2 (see [1,3,9,15]), etc.; surveys of various results about unit-distance graphs can be found at [10,11]. We also consider a similar notion of embeddability in Rd (see, e.g., [14]). A graph G = (V , E ) is embeddable in Rd if there exists a mapping ϕ : V → Rd such that ∥ϕ(u) − ϕ(v)∥Rd = 1 for all uv ∈ E. It is clear that any unit-distance graph in Rd is embeddable in Rd but the converse does not always hold. These two notions differ in the following:
• Different vertices of an embeddable graph may be assigned to the same point in Rd while all vertices of a unit-distance graph should be assigned to pairwise distinct points.
• Non-adjacent vertices of an embeddable graph can be located at distance 1 while non-adjacent vertices of a unit-distance graph are forbidden to be placed at distance 1. We will say that an embedding ϕ : V → Rd is strict if all non-adjacent pairs of vertices are not placed at distance 1; we will say that an embedding ϕ : V → Rd is injective if all vertices are mapped to pairwise distinct points of Rd . It is clear that a graph G is a unit-distance graph in Rd iff there exists a strict and injective embedding of G in Rd . Thus we obtain four different notions of embeddability (strict/non-strict, injective/non-injective) which include the two notions described above. For each of the four notions of embeddability in Rd we can pose the computational decision problem of determining embeddability of chosen type for the given graph; we shall call this problem Rd -UNIT-DISTANCE-(STRICT)-(INJECTIVE)EMBEDDABILITY depending on the embeddability type. The computational complexity of these problems is studied in [14,7]. In [7] it is shown that Rd -UNIT-DISTANCE-(STRICT)-(INJECTIVE)-EMBEDDABILITY is NP-hard for each type of
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2016.05.011 0012-365X/© 2016 Elsevier B.V. All rights reserved.
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Fig. 1. An example of a reduction graph embedded in R3 . Vertices of the auxiliary 1-simplex are in the center (not labeled), vertices of U and VG are constrained to a circle. Bold parts of the circle are ‘‘forbidden’’ for vertices of VG , three small non-bold arcs correspond to different colors of vertices of G. Chains constraining mutual position of vertices of VG and U are not shown.
embeddability and each value of d > 2. Unfortunately, the proof in [7] for the case d > 2 is false as it is based on the result [6], which was refuted by Raigorodskii in [12,13]. We provide more details on the previous proof in Section 5. In the present paper we provide a new proof for the case d > 2. The main result is Theorem 1. The computational problems Rd -UNIT-DISTANCE-EMBEDDABILITY, Rd -UNIT-DISTANCE-STRICT-EMBEDDABILITY, Rd -UNIT-DISTANCE-INJECTIVE-EMBEDDABILITY, Rd -UNIT-DISTANCE-STRICT-INJECTIVE-EMBEDDABILITY are NP-hard for each d > 2. To prove this result we construct a reduction of the classic NP-complete problem of graph 3-coloring (3-COLORING) (see [5]) to each of the four embeddability problems, which implies NP-hardness of all four mentioned problems. It should be mentioned that the question whether the described problems lie in NP is open. Outline of the reduction Here we briefly describe the following construction. We define a rod as a unit-distance graph which is ‘‘rigid’’ enough to sustain the same distance between a pair of its vertices in any embedding. We formalize the notion in Section 2. Lemma 1 states that, under certain requirements, a rod subgraph is interchangeable with a weighted edge of corresponding length in terms of embeddability of the whole graph. Consequently, if we can build a rod which sustains distance l between a pair of vertices, we can just as well use a weighted edge of length l in its place. It is clear that not all distances are realizable as the length of a rod (since there are only countably many rods). However, in Section 3 we show that the set of realizable distances is dense, and present a constructive way to build a rod such that its length is constrained to a chosen interval of non-zero length. Further in the construction, we use rods obtained this way as a kind of ‘‘wobbly’’ weighted edges since we have no way of setting the length precisely. We also show how to constrain distance between two vertices to a certain interval using a chain of two rods of suitable length. The main part of the reduction from 3-COLORING presented in Section 4 proceeds as follows. We constrain all vertices of the input graph G to a circle by connecting them to vertices of an auxiliary (d − 2)-simplex. Then, we introduce three special vertices u0 , u1 , u2 constrained to the circle along with chains between them so that they lie roughly at the vertices of an equilateral triangle inscribed in the circle. Further, we constrain all vertices of V so that they lie close to the middle of an arc formed by u0 , u1 , u2 (thus assigning a definite ‘‘color’’ to each vertex), and forbid for adjacent vertices to have the same ‘‘color’’ by subdividing edges of G into chains. To finally obtain a unit-distance graph, we replace the weighted edges by rods via Lemma 1 (for a brief illustration see Fig. 1). 2. The notion of a rod Let us introduce some necessary definitions. A weighted graph G = (V , E , w) is an ordered triple such that (V , E ) is a graph and w : E → R+ is a function that assigns a positive number to each element of E; for every edge e ∈ E we will say that w(e) is the length of the edge e. If w ≡ 1, the weighted graph G is called a unit-distance graph. A length embedding (or, more simply, an embedding) of the weighted graph G = (V , E , w) in Rd is a map ϕ : V → Rd such that for any edge e = uv ∈ E distance between ϕ(u) and ϕ(v) is equal to w(e).
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Remark. In the sequel, we will identify vertices of the graph with points of Rd that are their images under the embedding if that does not cause confusion. Let ϕ be an embedding of the weighted graph G = (V , E , w) in Rd . We will call the embedding ϕ generic if no three vertices of G lie on a (one-dimensional) straight line. An embedding is called non-critical if it is strict, injective and generic. Consider a weighted graph G = (V , E , w) and a pair of its vertices u, v ∈ V . The graph G is called a (d-dimensional) (u, v)-rod of length l if the following conditions hold:
• distance between vertices u and v is equal to l in each embedding of G in Rd ; • there exists a non-critical embedding of G in Rd . If a unit-distance graph G is also a d-dimensional (u, v)-rod of length l, we call G a d-dimensional unit distance (u, v)-rod of length l. A weighted graph G = (V , E , w) is called a (unit distance) d-dimensional rod of length l if there exist two vertices u, v ∈ V such that G is a (unit distance) d-dimensional (u, v)-rod of length l. We suppose that d > 2 is a fixed constant throughout the whole paper, thus in the sequel we will write ‘‘rod’’ for ‘‘ddimensional rod’’. The following lemma states that replacing an edge of a weighted graph with a rod of appropriate length does not affect embeddability of the graph as far as we are concerned. Lemma 1. Let G be a weighted graph, and e = uv be an edge of G with length l. Let H be a (u, v)-rod of length l, and suppose that VG ∩ VH = {u, v}. Let G′ be the graph obtained from G by removing e and adding vertices and edges of H, with edge lengths induced by corresponding length functions of G and H. Then:
• If there is no embedding of G in Rd , then there is no embedding of G′ in Rd . • If there exists a non-critical embedding of G in Rd , then there exists a non-critical embedding of G′ in Rd . Proof. In any embedding of G′ distance between vertices u and v is equal to l. Suppose we have an embedding of G′ ; we can erase all vertices outside VG to obtain an embedding of G. The first claim is thus proven. Now consider a non-critical embedding ϕG of the weighted graph G in Rd . Construct an embedding ϕG′ of G′ as follows:
• Let ϕG′ (x) = ϕG (x) for all x ∈ VG ; • Choose a non-critical embedding ϕH of the weighted graph H such that ϕH (u) = ϕG (u), ϕH (v) = ϕG (v) (such embedding exists since ∥ϕG (u) − ϕG (v)∥ = l and H is a (u, v)-rod of length l); let ϕG′ (y) = ϕH (y) for all y ∈ VH . It is clear that this definition of ϕG′ is consistent. However, it is possible that ϕG′ is not a non-critical embedding. Note that no vertex of VG′ \ {u, v} lies on the straight line uv since the embeddings ϕG and ϕH are non-critical. Let S denote the set of all rotations of Rd about the line uv . S is isomorphic to the (d − 2)-dimensional sphere (each rotation can be assigned to the image of some point which does not lie on the line uv ). For any ψ ∈ S let ψ ∗H ϕG′ denote the mapping from VG′ in Rd such that (ψ ∗H ϕG′ )(x) = ϕG′ (x) for every x ∈ VG and (ψ ∗H ϕG′ )(y) = ψ(ϕG′ (y)) for every y ∈ VH ; clearly, this definition is consistent. It is also clear that for every rotation ψ ∈ S the mapping ψ ∗H ϕG′ is an embedding of G′ in Rd . We now show that there exists a rotation ψ ∈ S such that ψ ∗H ϕG′ is a non-critical embedding of G′ in Rd . Consider all ψ ∈ S such that the embedding ψ ∗H ϕG′ is not non-critical for some reason. (a) The embedding ψ ∗H ϕG′ places two vertices of G′ (denote them by x and y) at the same point. It follows from the noncriticality of ϕG and ϕH that x and y cannot lie both in VG or both in VH . Thus without loss of generality x ∈ VG \ {u, v}, y ∈ VH \ {u, v}. The vertex y does not lie on the line uv and no two rotations place y at the same point. Therefore for every pair of vertices x, y there is at most one rotation ψ ∈ S that superposes x and ψ(y), thus the set of all rotations ψ such that the embedding ψ ∗H ϕG′ places some two vertices in the same point is finite and its spherical measure in S is zero. (b) The embedding ψ ∗H ϕG′ places two non-adjacent vertices of G′ (denote them by x and y once more) at distance 1. Once again, x, y ∈ VG or x, y ∈ VH leads to a contradiction; thus without loss of generality x ∈ VG \ {u, v}, y ∈ VH \ {u, v}. Let Py denote the (d − 2)-dimensional sphere that is the locus of the point ψ(y) as ψ ranges over S; the radius of Py is non-zero since y does not lie on the line uv . If ∥x − ψ(y)∥Rd = 1, then ψ(y) lies on the (d − 1)-dimensional sphere of radius 1 centered at x; denote it by Px . We assume that the intersection of Px and Py is not empty. If Px contains Py as a subset, then x must lie on the line uv ; that would contradict with non-criticality of ϕG . Otherwise, the intersection of Px and Py is a (d − 3)-dimensional sphere (possibly, of zero radius). Indeed, Px ∩ Py is the set of points of Rd obtained by fixing distances from three non-collinear points u, v , x. Denote the unique 2-plane passing through u, v , x by π . Consider any point p ∈ Px ∩ Py , denote by θ the (d − 2)-plane passing through p and orthogonal to π , and by O the point of intersection of π and θ . Using Pythagoras’ theorem, only can easily show that any point p′ ∈ Px ∩ Py lies in θ , and ∥p − O∥ = ∥p′ − O∥, moreover, Px ∩ Py is exactly the set of all such points. Thus, Px ∩ Py is a hypersphere in a (d − 2)-dimensional plane, that is, a (d − 3)-dimensional sphere. In any case, the set of rotations that place x and ψ(y) at distance 1 has zero measure in S. Thus the set of rotations that place some two non-adjacent vertices at distance 1 has zero measure in S.
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Fig. 2. 5-dimensional Moser spindle.
(c) The embedding ψ ∗H ϕG′ places some three vertices on a straight line; denote these vertices by x, y, z. Similarly to previous cases, if we assume x, y, z ∈ VG or VH we arrive at a contradiction. Suppose that exactly two vertices among x, y, z lie in VG . Without loss of generality, suppose that x, y ∈ VG , and z ∈ VH \ VG . Since the point z cannot lie on the line uv , the sphere {ψ(z ) : ψ ∈ S } has non-zero radius and the line xy passes through the point ψ(z ) for at most two values of ψ . Similarly, suppose that exactly two vertices among x, y, z lie in VH . Let x ∈ VG \ VH and y, z ∈ VH . The rotation ψ ∈ S places the point x on the line ψ(yz ) iff the point ψ −1 (x) lies on the line yz (here ψ −1 means the inverse rotation of ψ ), therefore in this case the line yz must cross the locus of ψ −1 (x) as ψ ranges over S. Clearly, the locus is a sphere of non-zero radius, thus the line ψ(yz ) passes through the point x for at most two values of ψ . It follows from the above that the set of rotations ψ ∈ S such that ψ ∗H ϕG′ places some three vertices on a straight line is finite. To sum up, the set of rotations ψ such that the embedding ψ ∗H ϕG′ is not non-critical has zero measure in the (d − 2)dimensional sphere of all possible rotations about the line uv . Therefore almost every rotation ψ ∈ S yields a non-critical embedding ψ ∗H ϕG′ of the graph G′ in Rd . 3. Construction of rods Let h =
d+1 2d
denote the altitude length of a regular d-simplex with the edge length 1; denote D = 2h. Clearly, D >
√ 2.
Lemma 2. Let G and H be unit distance rods of length a and b respectively. Then there exists a unit distance rod of length ab. Proof. It suffices to make lengths of all edges of G be equal to b and successively apply Lemma 1 to every edge of the resulting graph and the graph H. Consider a graph Md on a set of vertices VMd = K1 ∪ K2 ∪ {A, B, C }, |K1 | = |K2 | = d. Starting from the graph on VMd with no edges, add the following edges of unit length to Md : (a) (b) (c) (d)
make cliques on K1 and K2 ; connect the vertices A and B with every vertex of K1 ; connect the vertices A and C with every vertex of K2 ; finally, connect the vertices B and C .
The graph Md is called a d-dimensional Moser spindle (Fig. 2 illustrates a 5-dimensional Moser spindle). We claim that Md is a unit distance d-dimensional (A, B)-rod of length D. To see that, fix locations of vertices of K1 . Since there are exactly two possible positions for A and B, in any embedding B (and similarly, C ) either coincides with A or is at distance D from it. Since B and C are adjacent, it follows that in any embedding neither B nor C can coincide with A, therefore ∥A − B∥ = ∥A − C ∥ = D. Moreover, B and C can be placed at distance 1 since 2D > 1. To obtain a non-critical embedding, we can find suitable rotations of vertices of K1 around the line AB (similarly, vertices of K2 around the line AC ) using the same argument as in the proof of Lemma 1. Repeatedly applying Lemma 2 to copies of Md , we arrive at Corollary 1. For every non-negative integer k there exists a unit distance d-dimensional rod of length Dk .
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Lemma 3. For all numbers a, b such that 0 < a < b < 1 there exists a number l satisfying a < l < b and a graph G such that G is a unit-distance d-dimensional rod of length l. Proof. Construct G as follows. Choose a set of vertices K of size d − 1 and connect its elements pairwise by unit length edges. Then, take a sequence of vertices v0 , . . . , vn−1 (the exact number of vertices n > 2 will be determined later) and connect every vertex of the sequence vi with every vertex of K by a unit length edge. If the location of vertices of K is fixed, then all vertices v0 , . . . , vn−1 must lie on some circle centered at O, where O is the barycenter of the regular simplex with vertices in K . The radius of the circle is equal to the altitude length of a (d − 1)-simplex with side length 1, i.e.
d 2(d−1)
= r. Let π
denote the plane containing this circle. For each i from 0 to n − 2 connect the vertices vi and vi+1 by an edge of length 1; also for each i from 0 to n − 3 connect the vertices vi and vi+2 by an edge of length D. Now in every embedding of the graph G the angle ̸ vi Ovi+1 is equal to the dihedral angle of a regular d-simplex; denote this angle by α = arccos 1d . Additionally, the least rotation of the plane π about the point O that moves the point vi to vi+1 has the same direction for every i. It is clear that no three vertices of G lie on a straight line. Let us introduce an angular coordinate system ψ on π centered at O such that ψ(v0 ) = 0, ψ(v1 ) = α . Clearly, ψ(vj ) = jα mod 2π (by α mod 2π we mean α + k × 2π for an integer k such that 0 6 α + k × 2π < 2π ). By Niven’s theorem (see [8], Corollary 3.12), α/2π cannot be a rational number when d > 3, therefore the infinite sequence xj = jα mod 2π is b ) and ∥v0 − vN ∥ ∈ (a; b) in dense in [0; 2π ]. Thus there exists a positive integer N such that xN ∈ (2 arcsin 2ra ; 2 arcsin 2r any embedding of G. Put n = max(3, N + 1). Let us ensure that there is a non-critical embedding of the graph G. Denote by θ the (d − 2)-plane containing all vertices of K , and by π the 2-plane containing the vertices v0 , . . . , vn−1 . By a simple Pythagoras’ theorem argument we obtain that θ ⊥ π and θ ∩ π = {O}. Thus, any embedding of G is injective since no vertices vi , vj can be at the same point, no two vertices of K can be at the same point, and no vertex occupies the point O. Any embedding is also strict since vertices vi , vj with |i − j| ̸= 1 never lie at distance 1, and all other pairs share a unit-length edge. Let us ensure that any embedding of G is generic. Clearly, no three vertices of K lie on a straight line since they are vertices of a simplex. Further, no three vertices among v0 , . . . , vn−1 lie on a line since they lie on a circle. Finally, any 1-line not completely inside θ or π has at most one common point with each of them, therefore no other triple of points can occur on a 1-line. It follows from the above that the graph G is a d-dimensional (v0 , vN )-rod. Successively apply Lemma 2 to each D-length edge of the graph G and the graph Md ; the resulting graph is a unit distance d-dimensional (v1 , vN )-rod that satisfies all the conditions. Theorem 2. For all numbers a, b such that 0 < a < b there exists a number l satisfying a < l < b and a graph G such that G is a unit-distance d-dimensional rod of length l. Proof. Choose a non-negative integer k such that Dk > b and denote G′ the rod obtained by applying Lemma 3 for numbers a and Dbk . Now apply Lemma 2 to the graph G′ and the rod of length Dk . Dk Let RL(a, b) denote a number l produced by Theorem 2 for given numbers a and b, and Rod(a, b) denote the rod of the corresponding length. 4. The reduction setup Consider a graph G = (VG , EG ) which is the input of the 3-COLORING problem. We now construct a weighted graph H = (VH , EH , wH ) = 3-COLORING-Rd -EMBEDDABILITY-REDUCTION(G) such that the embeddability of H in Rd is equivalent to the existence of a solution to the 3-COLORING for the graph G. One operation we will heavily use in the construction is edge subdivision. Formally, we will say that subdividing an edge e = uv ∈ E of a graph (V , E ) results in the graph (V ∪ {e}, (E \ {e}) ∪ {v e, ue}). The following lemma is instrumental in manually introducing distance constraints in the weighted graph embedding. Lemma 4. Let 0 6 l < L 6 R < r be real numbers (possibly with r = +∞). Let G = (VG , EG , wG ) and H = (VH , EH , wH ) be weighted graphs, and u, v ∈ VG be distinct. Suppose that the graph (VH , EH ) is obtained from (VG , EG ) by adding the (possibly parallel) edge e = uv and immediately subdividing it. Suppose further that the weight function wH coincides with wG on EG \ {e}, and
wH (v e) = a ∈ (a, a),
wH (ue) = b ∈ (b, b),
where
δ = min(L − l, r − R), δ δ L+R L+R a= − , a= + , b=
2 R−L 2
3
δ
2 R−L
3
2
+ , b=
3
δ + . 2
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Then:
• In every embedding of the graph H the inequalities l < ∥v − u∥ < r hold. • If there exists a non-critical embedding of G such that L 6 ∥v − u∥ 6 R, then there exists a non-critical embedding of H. Proof. First of all, let us show that the inequalities l < a − b < L 6 R < a + b < r hold. Indeed, a−b>a−b= a−ba+b= a+b
L+R 2
L+R 2
L+R 2
L+R 2
− + − +
δ
3
δ
δ 3
R−L 2
+
R−L 2
+
3
R−L 2
−
3
δ
−
R−L 2
+ + + +
δ
2
δ
= L;
3
δ
= R;
3
δ
> L − δ > l;
2
< R + δ 6 r.
To prove the first part, consider any embedding of the graph H. It follows from the triangle inequality applied to vertices
v, e, u that l < a − b 6 ∥u − v∥ 6 a + b < r. The first claim is thus proven. Now, consider a non-critical embedding of the graph G such that ∥u − v∥ ∈ [L; R]. It follows from a − b < L 6 ∥u − v∥ 6 R < a + b that it is possible to place the vertex e in such a way that ∥v − e∥ = a, ∥u − e∥ = b and e does not lie on the line v u. We have obtained an embedding of the graph H. It is possible to obtain a non-critical embedding by choosing an appropriate rotation of e about the line v u; the proof of existence of such rotation copies the similar proof from Lemma 1 almost entirely.
Denote the altitude length of a (d − 1)-simplex with unit-length edges by r0 =
d , 2(d−1)
and the length of a chord that
π . joins the ends of an arc of length r0 α in a circle of radius r0 by Ch(α) = 2r0 sin α/2; also denote ε = 24 The weighted graph H is obtained by the following process. Start with the vertex set of the input graph G, vertices of the simplex K = {k1 , . . . , kd−1 }, and U = {u0 , u1 , u2 }. Make the subgraph induced by K complete, and connect vertices of V ∪ U with vertices of K pairwise. Denote the graph obtained at this point by H0 . Now, make the subgraph induced by U complete, connect each vertex of U with each vertex of V , and add edges of G. Finally, subdivide each edge induced by V ∪ U. The resulting graph is (VH , EH ). To assign weights to EH , first define
lU = Ch(2π /3 − ε/2), lUV = Ch(π /3 − ε), lV = Ch(5ε/2),
LU = Ch(2π /3),
RU = Ch(2π /3),
LUV = Ch(π /3 − ε/2), LV = Ch(2π /3 − ε),
rU = Ch(2π /3 + ε/2)
RUV = 2r0 , rUV = ∞
RV = 2r0 ,
rV = ∞.
(1)
Then, define δU and the interval endpoints aU , aU , bU , bU according to the statement of Lemma 4. Similarly, define analogous parameters with indices UV and V . For each subdivided edge e = ui uj with i < j, define
wH (ui e) = aU = RL(aU , aU ) and wH (euj ) = bU = RL(bU , bU ). For each subdivided edge e = ui vj with ui ∈ U, vj ∈ V , define
wH (ui e) = aUV = RL(aUV , aUV ) and wH (evj ) = bUV = RL(bUV , bUV ). For each subdivided edge e = vi vj ∈ EG with i < j, define
wH (vi e) = aV = RL(aV , aV ) and wH (evj ) = bV = RL(bV , bV ). For all other edges of H (namely, edges incident to at least one vertex of K ), we put wH to be 1. Theorem 3. Suppose G is the input graph and the weighted graph H = 3-COLORING-Rd -EMBEDDABILITY-REDUCTION(G) is constructed as described above. Then:
• If there is no proper 3-coloring of vertices of G, then there is no embedding of H in Rd . • If a proper 3-coloring of vertices of G exists, then there exists a non-critical embedding of H in Rd . Proof. Consider an embedding of the graph H in Rd . We will construct a proper 3-coloring of vertices of G. Start with the graph H0 . Each vertex of V ∪ U is at distance 1 from each vertex of the regular (d − 2)-simplex with unit side length induced by K , thus in every embedding of H0 all vertices of V ∪ U are located on some circle of radius r0 ; denote this circle by ρ and its center by O. Note that at this point each vertex of V ∪ U can be freely mapped to any point of ρ . Now successively add edges to H0 and subdivide them until we obtain H, and apply the first part of Lemma 4 on each subdivision.
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• Let us add and subdivide an edge e = ui uj , with wH (ui e) = aU and wH (euj ) = bU . Then, in each embedding Ch(2π / 3 − ε/2) = lU < ∥ui − uj ∥ < rU = Ch(2π /3 + ε/2), or, equivalently, 2π /3 − ε/2 < ̸ ui Ouj < 2π /3 + ε/2. • Let us add and subdivide an edge e = ui vj , with wH (ui e) = aUV and wH (evj ) = bUV . Then, Ch(π /3 − ε) = lUV < ∥ui − vj ∥, or π /3 − ε < ̸ ui Ovj . • Let us add and subdivide an edge e = vi vj , with wH (vi e) = aV and wH (evj ) = bV . Then, Ch(5ε/2) = lV < ∥vi − vj ∥, or 5ε/2 < ̸ vi Ovj . Construct the coloring of vertices of G as follows: if the vertex v ∈ VG lies on the shortest arc between u0 and u1 in the embedding of H, it is assigned to the color c (v) = 2; if v lies on the shortest arc between u0 and u2 , then c (v) = 1; otherwise, c (v) = 0. Clearly, this coloring is unambiguously defined for any embedding of H. Let us prove that this coloring of VG is proper, that is, for every edge vi vj ∈ E (G) we have c (vi ) ̸= c (vj ). Let us show that for every edge vi vj ∈ EG the shortest arc of ρ between the points vi and vj contains at least one vertex of U. Assume the contrary, then without loss of generality both vertices vi and vj lie on the shortest arc between u0 and u1 , and ̸ u0 Ou1 = ̸ u0 Ovi + ̸ vi Ovj + ̸ vj Ou1 > (π /3 − ε) + 52 ε + (π /3 − ε) = 2π /3 + ε/2. But that contradicts with ̸ u0 Ou1 < 2π /3 + ε/2, thus at least one vertex of U must lie between vi and vj . In that case c (vi ) ̸= c (vj ) therefore the coloring is proper. The first part of Theorem 3 is thus proven. Now consider a proper 3-coloring of G; let us construct a non-critical embedding of H. First, construct a non-critical embedding of H0 as follows. Choose an arbitrary regular (d − 2)-simplex with edge length 1 and identify its vertices with vertices of K . Let O denote the barycenter of the simplex and ρ denote the locus of all points at distance 1 from all vertices of the simplex. Clearly, ρ is a circle of radius r0 . Next, choose an arbitrary equilateral triangle inscribed in ρ and place the vertices u0 , u1 , u2 at the vertices of the triangle. Denote by γ0 the set of all points x ∈ ρ such that ̸ u0 Ox > π − ε/2. Clearly, γ0 is an open arc of angular measure ε; similarly define sets γ1 , γ2 ; Now, suppose that the vertex v ∈ VG is assigned to color c (v) ∈ {0, 1, 2} in the given 3-coloring. Then, place every vertex v ∈ V in such a way that v lies on the arc γc (v) , and no two vertices of V are at the same point. Since the arcs γ0 , γ1 , γ2 have non-zero angular measure, such arrangement of vertices of V is possible. Clearly, the arrangement of vertices of VH described above yields a non-critical embedding of the graph H0 . Now again, we will subsequently add-and-subdivide edges to H0 to obtain H. Each time we will apply the second part of Lemma 4 to justify the existence of a non-critical embedding.
• Let us add and subdivide an edge e = ui uj , with wH (ui e) = aU and wH (euj ) = bU . The points ui and uj are at the vertices of an equilateral triangle inscribed in the circle ρ , thus LU = ∥ui − uj ∥ = Ch(2π /3) = RU , and by Lemma 4 a non-critical embedding persists.
• Let us add and subdivide an edge e = ui vj , with wH (ui e) = aUV and wH (evj ) = bUV . There is at least one vertex uk ∈ U such that ̸ uk Ovj > π −ε/2, thus ̸ ui Ovj > |̸ uk Ovj − ̸ ui Ouk | > π /3 −ε/2, and LUV = Ch(π /3 −ε/2) < ∥ui −vj ∥ 6 2r0 = RUV , hence a non-critical embedding persists.
• Let us add and subdivide an edge e = vi vj , with wH (vi e) = aV and wH (evj ) = bV . The points vi and vj lie on different arcs γc (vi ) , γc (vj ) . Let uk denote a vertex of U that lies on the shortest arc between vi and vj . Then ̸ vi Ovj = ̸ vi Ouk + ̸ uk Ovj > 2(π /3 − ε/2) = 2π /3 − ε , and LV = Ch(2π /3 − ε) < ∥vi − vj ∥ 6 2r0 = RV , hence a non-critical embedding persists. In the end of the process of adding and subdividing edges we obtain a non-critical embedding of the graph H.
The constructed graph H has O(|VG | + |EG |) vertices and edges (we recall that the dimension d is a fixed constant), but it contains edges of non-unit length. However, every such edge has length RL(x, x) for some numbers x and x. Let K be the maximal size of Rod(x, x) over all used pairs (x, x). Observe that the set of pairs (x, x) is finite and independent of the input graph G (namely, (x, x) ∈ {(aU , aU ), (bU , bU ), . . .}), thus K = K (d) is a constant. Let us apply Lemma 2 to each non-unit length edge in order to replace it with a corresponding unit-length rod. The resulting graph H ′ is equivalently embeddable as H, and its size is at most K times the size of H. Therefore the resulting (non-weighted!) graph H ′ = 3-COLORING-Rd UNIT-DISTANCE-EMBEDDABILITY-REDUCTION(G) has O(|VG | + |EG |) vertices and edges as well. Finally, we obtain Theorem 4. Let the graph H ′ = 3-COLORING-Rd -UNIT-DISTANCE-EMBEDDABILITY-REDUCTION(G) be constructed by a given graph G = (VG , EG ) as described above. Then:
• If there is no proper 3-coloring of vertices of G, then there is no embedding of H ′ in Rd . • If a proper 3-coloring of vertices of G exists, then there exists a non-critical embedding of H ′ in Rd . From Theorem 4, the linearity of the size of H ′ , and the fact that the problem of 3-coloring is NP-hard (see [5]) Theorem 1 follows. 5. Discussion of the previous proof In this section we discuss the previous proof of the result found in [7], its dependence on the recently disproved result [6] and whether the flaw is critical. Here is an outline of the approach used in [7] to prove NP-hardness of the non-strict non-injective Rd -embeddability problem for any constant d > 2. Construct a reduction from d-COLORING to the embeddability problem. Build the graph Md
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in the same way as described in Section 3 (we adopt the same notation for vertices as in Section 3). Add the input graph G to the graph Md , and connect all its vertices to the vertices A and B. An assertion follows: the constructed graph H is embeddable in Rd iff χ (G) 6 d. This would imply correctness of the reduction, and NP-hardness would follow. It is trivial to obtain the leftwards implication: χ (G) 6 d implies that there is a homomorphism of G onto the clique subgraph induced by K1 , thus H is embeddable. The rightwards implication is based on the result of Lovász in [6] which is used in the following form: Theorem L (Lovász). Let d be a positive integer. Let Sd,r be the unit-distance graph of the (d − 1)-dimensional sphere with radius r. Then:
(b) Let r =
χ(
)
1 the inequality Sd,r > d holds. 2 d be the radius of the sphere circumscribed to the regular simplex with unit-length edges. Then, 2(d+1)
(a) For any r >
χ (Sd,r ) 6 d + 1.
Now, to obtain the rightwards implication, note that the (d − 2)-dimensional sphere S (K1 ) circumscribed to the vertices of K1 satisfies the second part of Theorem L, thus χ (S (K1 )) 6 d, which immediately implies χ (G) 6 d since vertices of G must lie on S (K1 ). While the argument itself is correct, the cited result in [6] turns out to be false. In [13] Raigorodskii refutes Theorem L and proves several lower bounds contrary to the upper bound Sd,r 6 d + 1. In particular, Theorem 6 of [13] implies that if the sequence rd is bounded away from 21 , then χ (Sd,rd ) > d + 1 for sufficiently large d, which is indeed our case since
d 2(d+1)
tends to √1 as d → ∞. Thus χ (S (K1 )) > d for large d, which renders the reduction invalid. 2
We note that the proof in [7] and the proof in the present paper share a general approach, which is to constrain the vertices of the COLORING input graph to some sphere (in the case of the present paper, a circle) and then argue about the chromatic number of the sphere. However, we suppose that it is hardly possible to establish a proper reduction this way if we do not introduce gadgets to better control distances and mutual arrangement of points, which is arguably one of the major contributions of the present paper. There are several possible ways to adjust the proof in [7], such as altering the dimension and/or radius of the sphere containing graph’s vertices (and indeed, Theorem 8 of [13] shows that a (d − 1)-dimensional sphere can have chromatic number at most d + 1 if its radius is sufficiently close to 1/2), but that would probably require much additional work as well as use of distance-controlling gadgets; this is even more so if we turn to ‘‘messier’’ (strict and/or injective) cases of embeddability. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P. Brass, W.O. Moser, J. Pach, Research Problems in Discrete Geometry, Vol. 18, Springer, 2005. K.B. Chilakamarri, The unit-distance graph problem: a brief survey and some new results, Bull. Inst. Combin. Appl. 8 (39) (1993) C60. N. de Bruijn, P. Erdős, A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. (N.S.) 13 (5) (1951) 371–373. P. Erdős, On sets of distances of n points, Amer. Math. Monthly (1946) 248–250. R.M. Karp, Reducibility Among Combinatorial Problems, Springer, 1972. L. Lovász, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. 45 (1–4) (1983) 317–323. B. Horvat, J. Kratochvíl, T. Pisanski, On the computational complexity of degenerate unit distance representations of graphs, in: Combinatorial Algorithms, Springer, 2011, pp. 274–285. I. Niven, Irrational numbers, carus math, in: Monographs, John Wiley and Sons Inc., 1956. A.M. Raigorodskii, Borsuk’s problem and the chromatic numbers of some metric spaces, Russian Math. Surveys 56 (1) (2001) 103–139. A.M. Raigorodskii, Coloring distance graphs and graphs of diameters, in: Thirty Essays on Geometric Graph Theory, Springer, 2013, pp. 429–460. A.M. Raigorodskii, Cliques and cycles in distance graphs and graphs of diameters, Discrete Geom. Algebr. Combin. 625 (2014) 93–110. A. Raigorodskii, On the chromatic numbers of spheres in euclidean spaces, in: Doklady Mathematics, Vol. 81, Springer, 2010, pp. 379–382. A. Raigorodskii, On the chromatic numbers of spheres in Rn , Combinatorica 32 (1) (2012) 111–123. J.B. Saxe, Embeddability of weighted graphs in k-space is strongly np-hard, in: Proc. 17th Allerton Conf. Commun. Control Comput, 1979, pp. 480–489. A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and The Colorful Life of Its Creators, Springer Science & Business Media, 2008.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
On computational complexity of length embeddability of graphs Mikhail Tikhomirov Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation
article
info
Article history: Received 21 October 2014 Received in revised form 11 May 2016 Accepted 11 May 2016
abstract A graph G is embeddable in Rd if the vertices of G can be assigned to points of Rd in such a way that all pairs of adjacent vertices are at distance 1. We show that verifying embeddability of a given graph in Rd is NP-hard in the case d > 2 for all reasonable notions of embeddability. © 2016 Elsevier B.V. All rights reserved.
Keywords: Computational complexity Graph embedding Distance graph
1. Introduction The unit-distance graph of S ⊂ Rd is defined as the graph G = (V , E ), where V = S and E is the set of all pairs of points x, y ∈ S such that x and y are at distance 1. A graph is a unit-distance graph in Rd if it is isomorphic to the unit-distance graph of some set S ⊂ Rd . Some famous problems concerning unit-distance graphs are the Erdős’ unit distance problem on the maximal number of unit distances between n points in R2 (see [1,4,2]), the Hadwiger–Nelson problem on the chromatic number of R2 (see [1,3,9,15]), etc.; surveys of various results about unit-distance graphs can be found at [10,11]. We also consider a similar notion of embeddability in Rd (see, e.g., [14]). A graph G = (V , E ) is embeddable in Rd if there exists a mapping ϕ : V → Rd such that ∥ϕ(u) − ϕ(v)∥Rd = 1 for all uv ∈ E. It is clear that any unit-distance graph in Rd is embeddable in Rd but the converse does not always hold. These two notions differ in the following:
• Different vertices of an embeddable graph may be assigned to the same point in Rd while all vertices of a unit-distance graph should be assigned to pairwise distinct points.
• Non-adjacent vertices of an embeddable graph can be located at distance 1 while non-adjacent vertices of a unit-distance graph are forbidden to be placed at distance 1. We will say that an embedding ϕ : V → Rd is strict if all non-adjacent pairs of vertices are not placed at distance 1; we will say that an embedding ϕ : V → Rd is injective if all vertices are mapped to pairwise distinct points of Rd . It is clear that a graph G is a unit-distance graph in Rd iff there exists a strict and injective embedding of G in Rd . Thus we obtain four different notions of embeddability (strict/non-strict, injective/non-injective) which include the two notions described above. For each of the four notions of embeddability in Rd we can pose the computational decision problem of determining embeddability of chosen type for the given graph; we shall call this problem Rd -UNIT-DISTANCE-(STRICT)-(INJECTIVE)EMBEDDABILITY depending on the embeddability type. The computational complexity of these problems is studied in [14,7]. In [7] it is shown that Rd -UNIT-DISTANCE-(STRICT)-(INJECTIVE)-EMBEDDABILITY is NP-hard for each type of
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2016.05.011 0012-365X/© 2016 Elsevier B.V. All rights reserved.
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Fig. 1. An example of a reduction graph embedded in R3 . Vertices of the auxiliary 1-simplex are in the center (not labeled), vertices of U and VG are constrained to a circle. Bold parts of the circle are ‘‘forbidden’’ for vertices of VG , three small non-bold arcs correspond to different colors of vertices of G. Chains constraining mutual position of vertices of VG and U are not shown.
embeddability and each value of d > 2. Unfortunately, the proof in [7] for the case d > 2 is false as it is based on the result [6], which was refuted by Raigorodskii in [12,13]. We provide more details on the previous proof in Section 5. In the present paper we provide a new proof for the case d > 2. The main result is Theorem 1. The computational problems Rd -UNIT-DISTANCE-EMBEDDABILITY, Rd -UNIT-DISTANCE-STRICT-EMBEDDABILITY, Rd -UNIT-DISTANCE-INJECTIVE-EMBEDDABILITY, Rd -UNIT-DISTANCE-STRICT-INJECTIVE-EMBEDDABILITY are NP-hard for each d > 2. To prove this result we construct a reduction of the classic NP-complete problem of graph 3-coloring (3-COLORING) (see [5]) to each of the four embeddability problems, which implies NP-hardness of all four mentioned problems. It should be mentioned that the question whether the described problems lie in NP is open. Outline of the reduction Here we briefly describe the following construction. We define a rod as a unit-distance graph which is ‘‘rigid’’ enough to sustain the same distance between a pair of its vertices in any embedding. We formalize the notion in Section 2. Lemma 1 states that, under certain requirements, a rod subgraph is interchangeable with a weighted edge of corresponding length in terms of embeddability of the whole graph. Consequently, if we can build a rod which sustains distance l between a pair of vertices, we can just as well use a weighted edge of length l in its place. It is clear that not all distances are realizable as the length of a rod (since there are only countably many rods). However, in Section 3 we show that the set of realizable distances is dense, and present a constructive way to build a rod such that its length is constrained to a chosen interval of non-zero length. Further in the construction, we use rods obtained this way as a kind of ‘‘wobbly’’ weighted edges since we have no way of setting the length precisely. We also show how to constrain distance between two vertices to a certain interval using a chain of two rods of suitable length. The main part of the reduction from 3-COLORING presented in Section 4 proceeds as follows. We constrain all vertices of the input graph G to a circle by connecting them to vertices of an auxiliary (d − 2)-simplex. Then, we introduce three special vertices u0 , u1 , u2 constrained to the circle along with chains between them so that they lie roughly at the vertices of an equilateral triangle inscribed in the circle. Further, we constrain all vertices of V so that they lie close to the middle of an arc formed by u0 , u1 , u2 (thus assigning a definite ‘‘color’’ to each vertex), and forbid for adjacent vertices to have the same ‘‘color’’ by subdividing edges of G into chains. To finally obtain a unit-distance graph, we replace the weighted edges by rods via Lemma 1 (for a brief illustration see Fig. 1). 2. The notion of a rod Let us introduce some necessary definitions. A weighted graph G = (V , E , w) is an ordered triple such that (V , E ) is a graph and w : E → R+ is a function that assigns a positive number to each element of E; for every edge e ∈ E we will say that w(e) is the length of the edge e. If w ≡ 1, the weighted graph G is called a unit-distance graph. A length embedding (or, more simply, an embedding) of the weighted graph G = (V , E , w) in Rd is a map ϕ : V → Rd such that for any edge e = uv ∈ E distance between ϕ(u) and ϕ(v) is equal to w(e).
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Remark. In the sequel, we will identify vertices of the graph with points of Rd that are their images under the embedding if that does not cause confusion. Let ϕ be an embedding of the weighted graph G = (V , E , w) in Rd . We will call the embedding ϕ generic if no three vertices of G lie on a (one-dimensional) straight line. An embedding is called non-critical if it is strict, injective and generic. Consider a weighted graph G = (V , E , w) and a pair of its vertices u, v ∈ V . The graph G is called a (d-dimensional) (u, v)-rod of length l if the following conditions hold:
• distance between vertices u and v is equal to l in each embedding of G in Rd ; • there exists a non-critical embedding of G in Rd . If a unit-distance graph G is also a d-dimensional (u, v)-rod of length l, we call G a d-dimensional unit distance (u, v)-rod of length l. A weighted graph G = (V , E , w) is called a (unit distance) d-dimensional rod of length l if there exist two vertices u, v ∈ V such that G is a (unit distance) d-dimensional (u, v)-rod of length l. We suppose that d > 2 is a fixed constant throughout the whole paper, thus in the sequel we will write ‘‘rod’’ for ‘‘ddimensional rod’’. The following lemma states that replacing an edge of a weighted graph with a rod of appropriate length does not affect embeddability of the graph as far as we are concerned. Lemma 1. Let G be a weighted graph, and e = uv be an edge of G with length l. Let H be a (u, v)-rod of length l, and suppose that VG ∩ VH = {u, v}. Let G′ be the graph obtained from G by removing e and adding vertices and edges of H, with edge lengths induced by corresponding length functions of G and H. Then:
• If there is no embedding of G in Rd , then there is no embedding of G′ in Rd . • If there exists a non-critical embedding of G in Rd , then there exists a non-critical embedding of G′ in Rd . Proof. In any embedding of G′ distance between vertices u and v is equal to l. Suppose we have an embedding of G′ ; we can erase all vertices outside VG to obtain an embedding of G. The first claim is thus proven. Now consider a non-critical embedding ϕG of the weighted graph G in Rd . Construct an embedding ϕG′ of G′ as follows:
• Let ϕG′ (x) = ϕG (x) for all x ∈ VG ; • Choose a non-critical embedding ϕH of the weighted graph H such that ϕH (u) = ϕG (u), ϕH (v) = ϕG (v) (such embedding exists since ∥ϕG (u) − ϕG (v)∥ = l and H is a (u, v)-rod of length l); let ϕG′ (y) = ϕH (y) for all y ∈ VH . It is clear that this definition of ϕG′ is consistent. However, it is possible that ϕG′ is not a non-critical embedding. Note that no vertex of VG′ \ {u, v} lies on the straight line uv since the embeddings ϕG and ϕH are non-critical. Let S denote the set of all rotations of Rd about the line uv . S is isomorphic to the (d − 2)-dimensional sphere (each rotation can be assigned to the image of some point which does not lie on the line uv ). For any ψ ∈ S let ψ ∗H ϕG′ denote the mapping from VG′ in Rd such that (ψ ∗H ϕG′ )(x) = ϕG′ (x) for every x ∈ VG and (ψ ∗H ϕG′ )(y) = ψ(ϕG′ (y)) for every y ∈ VH ; clearly, this definition is consistent. It is also clear that for every rotation ψ ∈ S the mapping ψ ∗H ϕG′ is an embedding of G′ in Rd . We now show that there exists a rotation ψ ∈ S such that ψ ∗H ϕG′ is a non-critical embedding of G′ in Rd . Consider all ψ ∈ S such that the embedding ψ ∗H ϕG′ is not non-critical for some reason. (a) The embedding ψ ∗H ϕG′ places two vertices of G′ (denote them by x and y) at the same point. It follows from the noncriticality of ϕG and ϕH that x and y cannot lie both in VG or both in VH . Thus without loss of generality x ∈ VG \ {u, v}, y ∈ VH \ {u, v}. The vertex y does not lie on the line uv and no two rotations place y at the same point. Therefore for every pair of vertices x, y there is at most one rotation ψ ∈ S that superposes x and ψ(y), thus the set of all rotations ψ such that the embedding ψ ∗H ϕG′ places some two vertices in the same point is finite and its spherical measure in S is zero. (b) The embedding ψ ∗H ϕG′ places two non-adjacent vertices of G′ (denote them by x and y once more) at distance 1. Once again, x, y ∈ VG or x, y ∈ VH leads to a contradiction; thus without loss of generality x ∈ VG \ {u, v}, y ∈ VH \ {u, v}. Let Py denote the (d − 2)-dimensional sphere that is the locus of the point ψ(y) as ψ ranges over S; the radius of Py is non-zero since y does not lie on the line uv . If ∥x − ψ(y)∥Rd = 1, then ψ(y) lies on the (d − 1)-dimensional sphere of radius 1 centered at x; denote it by Px . We assume that the intersection of Px and Py is not empty. If Px contains Py as a subset, then x must lie on the line uv ; that would contradict with non-criticality of ϕG . Otherwise, the intersection of Px and Py is a (d − 3)-dimensional sphere (possibly, of zero radius). Indeed, Px ∩ Py is the set of points of Rd obtained by fixing distances from three non-collinear points u, v , x. Denote the unique 2-plane passing through u, v , x by π . Consider any point p ∈ Px ∩ Py , denote by θ the (d − 2)-plane passing through p and orthogonal to π , and by O the point of intersection of π and θ . Using Pythagoras’ theorem, only can easily show that any point p′ ∈ Px ∩ Py lies in θ , and ∥p − O∥ = ∥p′ − O∥, moreover, Px ∩ Py is exactly the set of all such points. Thus, Px ∩ Py is a hypersphere in a (d − 2)-dimensional plane, that is, a (d − 3)-dimensional sphere. In any case, the set of rotations that place x and ψ(y) at distance 1 has zero measure in S. Thus the set of rotations that place some two non-adjacent vertices at distance 1 has zero measure in S.
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Fig. 2. 5-dimensional Moser spindle.
(c) The embedding ψ ∗H ϕG′ places some three vertices on a straight line; denote these vertices by x, y, z. Similarly to previous cases, if we assume x, y, z ∈ VG or VH we arrive at a contradiction. Suppose that exactly two vertices among x, y, z lie in VG . Without loss of generality, suppose that x, y ∈ VG , and z ∈ VH \ VG . Since the point z cannot lie on the line uv , the sphere {ψ(z ) : ψ ∈ S } has non-zero radius and the line xy passes through the point ψ(z ) for at most two values of ψ . Similarly, suppose that exactly two vertices among x, y, z lie in VH . Let x ∈ VG \ VH and y, z ∈ VH . The rotation ψ ∈ S places the point x on the line ψ(yz ) iff the point ψ −1 (x) lies on the line yz (here ψ −1 means the inverse rotation of ψ ), therefore in this case the line yz must cross the locus of ψ −1 (x) as ψ ranges over S. Clearly, the locus is a sphere of non-zero radius, thus the line ψ(yz ) passes through the point x for at most two values of ψ . It follows from the above that the set of rotations ψ ∈ S such that ψ ∗H ϕG′ places some three vertices on a straight line is finite. To sum up, the set of rotations ψ such that the embedding ψ ∗H ϕG′ is not non-critical has zero measure in the (d − 2)dimensional sphere of all possible rotations about the line uv . Therefore almost every rotation ψ ∈ S yields a non-critical embedding ψ ∗H ϕG′ of the graph G′ in Rd . 3. Construction of rods Let h =
d+1 2d
denote the altitude length of a regular d-simplex with the edge length 1; denote D = 2h. Clearly, D >
√ 2.
Lemma 2. Let G and H be unit distance rods of length a and b respectively. Then there exists a unit distance rod of length ab. Proof. It suffices to make lengths of all edges of G be equal to b and successively apply Lemma 1 to every edge of the resulting graph and the graph H. Consider a graph Md on a set of vertices VMd = K1 ∪ K2 ∪ {A, B, C }, |K1 | = |K2 | = d. Starting from the graph on VMd with no edges, add the following edges of unit length to Md : (a) (b) (c) (d)
make cliques on K1 and K2 ; connect the vertices A and B with every vertex of K1 ; connect the vertices A and C with every vertex of K2 ; finally, connect the vertices B and C .
The graph Md is called a d-dimensional Moser spindle (Fig. 2 illustrates a 5-dimensional Moser spindle). We claim that Md is a unit distance d-dimensional (A, B)-rod of length D. To see that, fix locations of vertices of K1 . Since there are exactly two possible positions for A and B, in any embedding B (and similarly, C ) either coincides with A or is at distance D from it. Since B and C are adjacent, it follows that in any embedding neither B nor C can coincide with A, therefore ∥A − B∥ = ∥A − C ∥ = D. Moreover, B and C can be placed at distance 1 since 2D > 1. To obtain a non-critical embedding, we can find suitable rotations of vertices of K1 around the line AB (similarly, vertices of K2 around the line AC ) using the same argument as in the proof of Lemma 1. Repeatedly applying Lemma 2 to copies of Md , we arrive at Corollary 1. For every non-negative integer k there exists a unit distance d-dimensional rod of length Dk .
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Lemma 3. For all numbers a, b such that 0 < a < b < 1 there exists a number l satisfying a < l < b and a graph G such that G is a unit-distance d-dimensional rod of length l. Proof. Construct G as follows. Choose a set of vertices K of size d − 1 and connect its elements pairwise by unit length edges. Then, take a sequence of vertices v0 , . . . , vn−1 (the exact number of vertices n > 2 will be determined later) and connect every vertex of the sequence vi with every vertex of K by a unit length edge. If the location of vertices of K is fixed, then all vertices v0 , . . . , vn−1 must lie on some circle centered at O, where O is the barycenter of the regular simplex with vertices in K . The radius of the circle is equal to the altitude length of a (d − 1)-simplex with side length 1, i.e.
d 2(d−1)
= r. Let π
denote the plane containing this circle. For each i from 0 to n − 2 connect the vertices vi and vi+1 by an edge of length 1; also for each i from 0 to n − 3 connect the vertices vi and vi+2 by an edge of length D. Now in every embedding of the graph G the angle ̸ vi Ovi+1 is equal to the dihedral angle of a regular d-simplex; denote this angle by α = arccos 1d . Additionally, the least rotation of the plane π about the point O that moves the point vi to vi+1 has the same direction for every i. It is clear that no three vertices of G lie on a straight line. Let us introduce an angular coordinate system ψ on π centered at O such that ψ(v0 ) = 0, ψ(v1 ) = α . Clearly, ψ(vj ) = jα mod 2π (by α mod 2π we mean α + k × 2π for an integer k such that 0 6 α + k × 2π < 2π ). By Niven’s theorem (see [8], Corollary 3.12), α/2π cannot be a rational number when d > 3, therefore the infinite sequence xj = jα mod 2π is b ) and ∥v0 − vN ∥ ∈ (a; b) in dense in [0; 2π ]. Thus there exists a positive integer N such that xN ∈ (2 arcsin 2ra ; 2 arcsin 2r any embedding of G. Put n = max(3, N + 1). Let us ensure that there is a non-critical embedding of the graph G. Denote by θ the (d − 2)-plane containing all vertices of K , and by π the 2-plane containing the vertices v0 , . . . , vn−1 . By a simple Pythagoras’ theorem argument we obtain that θ ⊥ π and θ ∩ π = {O}. Thus, any embedding of G is injective since no vertices vi , vj can be at the same point, no two vertices of K can be at the same point, and no vertex occupies the point O. Any embedding is also strict since vertices vi , vj with |i − j| ̸= 1 never lie at distance 1, and all other pairs share a unit-length edge. Let us ensure that any embedding of G is generic. Clearly, no three vertices of K lie on a straight line since they are vertices of a simplex. Further, no three vertices among v0 , . . . , vn−1 lie on a line since they lie on a circle. Finally, any 1-line not completely inside θ or π has at most one common point with each of them, therefore no other triple of points can occur on a 1-line. It follows from the above that the graph G is a d-dimensional (v0 , vN )-rod. Successively apply Lemma 2 to each D-length edge of the graph G and the graph Md ; the resulting graph is a unit distance d-dimensional (v1 , vN )-rod that satisfies all the conditions. Theorem 2. For all numbers a, b such that 0 < a < b there exists a number l satisfying a < l < b and a graph G such that G is a unit-distance d-dimensional rod of length l. Proof. Choose a non-negative integer k such that Dk > b and denote G′ the rod obtained by applying Lemma 3 for numbers a and Dbk . Now apply Lemma 2 to the graph G′ and the rod of length Dk . Dk Let RL(a, b) denote a number l produced by Theorem 2 for given numbers a and b, and Rod(a, b) denote the rod of the corresponding length. 4. The reduction setup Consider a graph G = (VG , EG ) which is the input of the 3-COLORING problem. We now construct a weighted graph H = (VH , EH , wH ) = 3-COLORING-Rd -EMBEDDABILITY-REDUCTION(G) such that the embeddability of H in Rd is equivalent to the existence of a solution to the 3-COLORING for the graph G. One operation we will heavily use in the construction is edge subdivision. Formally, we will say that subdividing an edge e = uv ∈ E of a graph (V , E ) results in the graph (V ∪ {e}, (E \ {e}) ∪ {v e, ue}). The following lemma is instrumental in manually introducing distance constraints in the weighted graph embedding. Lemma 4. Let 0 6 l < L 6 R < r be real numbers (possibly with r = +∞). Let G = (VG , EG , wG ) and H = (VH , EH , wH ) be weighted graphs, and u, v ∈ VG be distinct. Suppose that the graph (VH , EH ) is obtained from (VG , EG ) by adding the (possibly parallel) edge e = uv and immediately subdividing it. Suppose further that the weight function wH coincides with wG on EG \ {e}, and
wH (v e) = a ∈ (a, a),
wH (ue) = b ∈ (b, b),
where
δ = min(L − l, r − R), δ δ L+R L+R a= − , a= + , b=
2 R−L 2
3
δ
2 R−L
3
2
+ , b=
3
δ + . 2
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Then:
• In every embedding of the graph H the inequalities l < ∥v − u∥ < r hold. • If there exists a non-critical embedding of G such that L 6 ∥v − u∥ 6 R, then there exists a non-critical embedding of H. Proof. First of all, let us show that the inequalities l < a − b < L 6 R < a + b < r hold. Indeed, a−b>a−b= a−b
L+R 2
L+R 2
L+R 2
L+R 2
− + − +
δ
3
δ
δ 3
R−L 2
+
R−L 2
+
3
R−L 2
−
3
δ
−
R−L 2
+ + + +
δ
2
δ
= L;
3
δ
= R;
3
δ
> L − δ > l;
2
< R + δ 6 r.
To prove the first part, consider any embedding of the graph H. It follows from the triangle inequality applied to vertices
v, e, u that l < a − b 6 ∥u − v∥ 6 a + b < r. The first claim is thus proven. Now, consider a non-critical embedding of the graph G such that ∥u − v∥ ∈ [L; R]. It follows from a − b < L 6 ∥u − v∥ 6 R < a + b that it is possible to place the vertex e in such a way that ∥v − e∥ = a, ∥u − e∥ = b and e does not lie on the line v u. We have obtained an embedding of the graph H. It is possible to obtain a non-critical embedding by choosing an appropriate rotation of e about the line v u; the proof of existence of such rotation copies the similar proof from Lemma 1 almost entirely.
Denote the altitude length of a (d − 1)-simplex with unit-length edges by r0 =
d , 2(d−1)
and the length of a chord that
π . joins the ends of an arc of length r0 α in a circle of radius r0 by Ch(α) = 2r0 sin α/2; also denote ε = 24 The weighted graph H is obtained by the following process. Start with the vertex set of the input graph G, vertices of the simplex K = {k1 , . . . , kd−1 }, and U = {u0 , u1 , u2 }. Make the subgraph induced by K complete, and connect vertices of V ∪ U with vertices of K pairwise. Denote the graph obtained at this point by H0 . Now, make the subgraph induced by U complete, connect each vertex of U with each vertex of V , and add edges of G. Finally, subdivide each edge induced by V ∪ U. The resulting graph is (VH , EH ). To assign weights to EH , first define
lU = Ch(2π /3 − ε/2), lUV = Ch(π /3 − ε), lV = Ch(5ε/2),
LU = Ch(2π /3),
RU = Ch(2π /3),
LUV = Ch(π /3 − ε/2), LV = Ch(2π /3 − ε),
rU = Ch(2π /3 + ε/2)
RUV = 2r0 , rUV = ∞
RV = 2r0 ,
rV = ∞.
(1)
Then, define δU and the interval endpoints aU , aU , bU , bU according to the statement of Lemma 4. Similarly, define analogous parameters with indices UV and V . For each subdivided edge e = ui uj with i < j, define
wH (ui e) = aU = RL(aU , aU ) and wH (euj ) = bU = RL(bU , bU ). For each subdivided edge e = ui vj with ui ∈ U, vj ∈ V , define
wH (ui e) = aUV = RL(aUV , aUV ) and wH (evj ) = bUV = RL(bUV , bUV ). For each subdivided edge e = vi vj ∈ EG with i < j, define
wH (vi e) = aV = RL(aV , aV ) and wH (evj ) = bV = RL(bV , bV ). For all other edges of H (namely, edges incident to at least one vertex of K ), we put wH to be 1. Theorem 3. Suppose G is the input graph and the weighted graph H = 3-COLORING-Rd -EMBEDDABILITY-REDUCTION(G) is constructed as described above. Then:
• If there is no proper 3-coloring of vertices of G, then there is no embedding of H in Rd . • If a proper 3-coloring of vertices of G exists, then there exists a non-critical embedding of H in Rd . Proof. Consider an embedding of the graph H in Rd . We will construct a proper 3-coloring of vertices of G. Start with the graph H0 . Each vertex of V ∪ U is at distance 1 from each vertex of the regular (d − 2)-simplex with unit side length induced by K , thus in every embedding of H0 all vertices of V ∪ U are located on some circle of radius r0 ; denote this circle by ρ and its center by O. Note that at this point each vertex of V ∪ U can be freely mapped to any point of ρ . Now successively add edges to H0 and subdivide them until we obtain H, and apply the first part of Lemma 4 on each subdivision.
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• Let us add and subdivide an edge e = ui uj , with wH (ui e) = aU and wH (euj ) = bU . Then, in each embedding Ch(2π / 3 − ε/2) = lU < ∥ui − uj ∥ < rU = Ch(2π /3 + ε/2), or, equivalently, 2π /3 − ε/2 < ̸ ui Ouj < 2π /3 + ε/2. • Let us add and subdivide an edge e = ui vj , with wH (ui e) = aUV and wH (evj ) = bUV . Then, Ch(π /3 − ε) = lUV < ∥ui − vj ∥, or π /3 − ε < ̸ ui Ovj . • Let us add and subdivide an edge e = vi vj , with wH (vi e) = aV and wH (evj ) = bV . Then, Ch(5ε/2) = lV < ∥vi − vj ∥, or 5ε/2 < ̸ vi Ovj . Construct the coloring of vertices of G as follows: if the vertex v ∈ VG lies on the shortest arc between u0 and u1 in the embedding of H, it is assigned to the color c (v) = 2; if v lies on the shortest arc between u0 and u2 , then c (v) = 1; otherwise, c (v) = 0. Clearly, this coloring is unambiguously defined for any embedding of H. Let us prove that this coloring of VG is proper, that is, for every edge vi vj ∈ E (G) we have c (vi ) ̸= c (vj ). Let us show that for every edge vi vj ∈ EG the shortest arc of ρ between the points vi and vj contains at least one vertex of U. Assume the contrary, then without loss of generality both vertices vi and vj lie on the shortest arc between u0 and u1 , and ̸ u0 Ou1 = ̸ u0 Ovi + ̸ vi Ovj + ̸ vj Ou1 > (π /3 − ε) + 52 ε + (π /3 − ε) = 2π /3 + ε/2. But that contradicts with ̸ u0 Ou1 < 2π /3 + ε/2, thus at least one vertex of U must lie between vi and vj . In that case c (vi ) ̸= c (vj ) therefore the coloring is proper. The first part of Theorem 3 is thus proven. Now consider a proper 3-coloring of G; let us construct a non-critical embedding of H. First, construct a non-critical embedding of H0 as follows. Choose an arbitrary regular (d − 2)-simplex with edge length 1 and identify its vertices with vertices of K . Let O denote the barycenter of the simplex and ρ denote the locus of all points at distance 1 from all vertices of the simplex. Clearly, ρ is a circle of radius r0 . Next, choose an arbitrary equilateral triangle inscribed in ρ and place the vertices u0 , u1 , u2 at the vertices of the triangle. Denote by γ0 the set of all points x ∈ ρ such that ̸ u0 Ox > π − ε/2. Clearly, γ0 is an open arc of angular measure ε; similarly define sets γ1 , γ2 ; Now, suppose that the vertex v ∈ VG is assigned to color c (v) ∈ {0, 1, 2} in the given 3-coloring. Then, place every vertex v ∈ V in such a way that v lies on the arc γc (v) , and no two vertices of V are at the same point. Since the arcs γ0 , γ1 , γ2 have non-zero angular measure, such arrangement of vertices of V is possible. Clearly, the arrangement of vertices of VH described above yields a non-critical embedding of the graph H0 . Now again, we will subsequently add-and-subdivide edges to H0 to obtain H. Each time we will apply the second part of Lemma 4 to justify the existence of a non-critical embedding.
• Let us add and subdivide an edge e = ui uj , with wH (ui e) = aU and wH (euj ) = bU . The points ui and uj are at the vertices of an equilateral triangle inscribed in the circle ρ , thus LU = ∥ui − uj ∥ = Ch(2π /3) = RU , and by Lemma 4 a non-critical embedding persists.
• Let us add and subdivide an edge e = ui vj , with wH (ui e) = aUV and wH (evj ) = bUV . There is at least one vertex uk ∈ U such that ̸ uk Ovj > π −ε/2, thus ̸ ui Ovj > |̸ uk Ovj − ̸ ui Ouk | > π /3 −ε/2, and LUV = Ch(π /3 −ε/2) < ∥ui −vj ∥ 6 2r0 = RUV , hence a non-critical embedding persists.
• Let us add and subdivide an edge e = vi vj , with wH (vi e) = aV and wH (evj ) = bV . The points vi and vj lie on different arcs γc (vi ) , γc (vj ) . Let uk denote a vertex of U that lies on the shortest arc between vi and vj . Then ̸ vi Ovj = ̸ vi Ouk + ̸ uk Ovj > 2(π /3 − ε/2) = 2π /3 − ε , and LV = Ch(2π /3 − ε) < ∥vi − vj ∥ 6 2r0 = RV , hence a non-critical embedding persists. In the end of the process of adding and subdividing edges we obtain a non-critical embedding of the graph H.
The constructed graph H has O(|VG | + |EG |) vertices and edges (we recall that the dimension d is a fixed constant), but it contains edges of non-unit length. However, every such edge has length RL(x, x) for some numbers x and x. Let K be the maximal size of Rod(x, x) over all used pairs (x, x). Observe that the set of pairs (x, x) is finite and independent of the input graph G (namely, (x, x) ∈ {(aU , aU ), (bU , bU ), . . .}), thus K = K (d) is a constant. Let us apply Lemma 2 to each non-unit length edge in order to replace it with a corresponding unit-length rod. The resulting graph H ′ is equivalently embeddable as H, and its size is at most K times the size of H. Therefore the resulting (non-weighted!) graph H ′ = 3-COLORING-Rd UNIT-DISTANCE-EMBEDDABILITY-REDUCTION(G) has O(|VG | + |EG |) vertices and edges as well. Finally, we obtain Theorem 4. Let the graph H ′ = 3-COLORING-Rd -UNIT-DISTANCE-EMBEDDABILITY-REDUCTION(G) be constructed by a given graph G = (VG , EG ) as described above. Then:
• If there is no proper 3-coloring of vertices of G, then there is no embedding of H ′ in Rd . • If a proper 3-coloring of vertices of G exists, then there exists a non-critical embedding of H ′ in Rd . From Theorem 4, the linearity of the size of H ′ , and the fact that the problem of 3-coloring is NP-hard (see [5]) Theorem 1 follows. 5. Discussion of the previous proof In this section we discuss the previous proof of the result found in [7], its dependence on the recently disproved result [6] and whether the flaw is critical. Here is an outline of the approach used in [7] to prove NP-hardness of the non-strict non-injective Rd -embeddability problem for any constant d > 2. Construct a reduction from d-COLORING to the embeddability problem. Build the graph Md
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M. Tikhomirov / Discrete Mathematics 339 (2016) 2605–2612
in the same way as described in Section 3 (we adopt the same notation for vertices as in Section 3). Add the input graph G to the graph Md , and connect all its vertices to the vertices A and B. An assertion follows: the constructed graph H is embeddable in Rd iff χ (G) 6 d. This would imply correctness of the reduction, and NP-hardness would follow. It is trivial to obtain the leftwards implication: χ (G) 6 d implies that there is a homomorphism of G onto the clique subgraph induced by K1 , thus H is embeddable. The rightwards implication is based on the result of Lovász in [6] which is used in the following form: Theorem L (Lovász). Let d be a positive integer. Let Sd,r be the unit-distance graph of the (d − 1)-dimensional sphere with radius r. Then:
(b) Let r =
χ(
)
1 the inequality Sd,r > d holds. 2 d be the radius of the sphere circumscribed to the regular simplex with unit-length edges. Then, 2(d+1)
(a) For any r >
χ (Sd,r ) 6 d + 1.
Now, to obtain the rightwards implication, note that the (d − 2)-dimensional sphere S (K1 ) circumscribed to the vertices of K1 satisfies the second part of Theorem L, thus χ (S (K1 )) 6 d, which immediately implies χ (G) 6 d since vertices of G must lie on S (K1 ). While the argument itself is correct, the cited result in [6] turns out to be false. In [13] Raigorodskii refutes Theorem L and proves several lower bounds contrary to the upper bound Sd,r 6 d + 1. In particular, Theorem 6 of [13] implies that if the sequence rd is bounded away from 21 , then χ (Sd,rd ) > d + 1 for sufficiently large d, which is indeed our case since
d 2(d+1)
tends to √1 as d → ∞. Thus χ (S (K1 )) > d for large d, which renders the reduction invalid. 2
We note that the proof in [7] and the proof in the present paper share a general approach, which is to constrain the vertices of the COLORING input graph to some sphere (in the case of the present paper, a circle) and then argue about the chromatic number of the sphere. However, we suppose that it is hardly possible to establish a proper reduction this way if we do not introduce gadgets to better control distances and mutual arrangement of points, which is arguably one of the major contributions of the present paper. There are several possible ways to adjust the proof in [7], such as altering the dimension and/or radius of the sphere containing graph’s vertices (and indeed, Theorem 8 of [13] shows that a (d − 1)-dimensional sphere can have chromatic number at most d + 1 if its radius is sufficiently close to 1/2), but that would probably require much additional work as well as use of distance-controlling gadgets; this is even more so if we turn to ‘‘messier’’ (strict and/or injective) cases of embeddability. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P. Brass, W.O. Moser, J. Pach, Research Problems in Discrete Geometry, Vol. 18, Springer, 2005. K.B. Chilakamarri, The unit-distance graph problem: a brief survey and some new results, Bull. Inst. Combin. Appl. 8 (39) (1993) C60. N. de Bruijn, P. Erdős, A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. (N.S.) 13 (5) (1951) 371–373. P. Erdős, On sets of distances of n points, Amer. Math. Monthly (1946) 248–250. R.M. Karp, Reducibility Among Combinatorial Problems, Springer, 1972. L. Lovász, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. 45 (1–4) (1983) 317–323. B. Horvat, J. Kratochvíl, T. Pisanski, On the computational complexity of degenerate unit distance representations of graphs, in: Combinatorial Algorithms, Springer, 2011, pp. 274–285. I. Niven, Irrational numbers, carus math, in: Monographs, John Wiley and Sons Inc., 1956. A.M. Raigorodskii, Borsuk’s problem and the chromatic numbers of some metric spaces, Russian Math. Surveys 56 (1) (2001) 103–139. A.M. Raigorodskii, Coloring distance graphs and graphs of diameters, in: Thirty Essays on Geometric Graph Theory, Springer, 2013, pp. 429–460. A.M. Raigorodskii, Cliques and cycles in distance graphs and graphs of diameters, Discrete Geom. Algebr. Combin. 625 (2014) 93–110. A. Raigorodskii, On the chromatic numbers of spheres in euclidean spaces, in: Doklady Mathematics, Vol. 81, Springer, 2010, pp. 379–382. A. Raigorodskii, On the chromatic numbers of spheres in Rn , Combinatorica 32 (1) (2012) 111–123. J.B. Saxe, Embeddability of weighted graphs in k-space is strongly np-hard, in: Proc. 17th Allerton Conf. Commun. Control Comput, 1979, pp. 480–489. A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and The Colorful Life of Its Creators, Springer Science & Business Media, 2008.