Posets and planar graphs

POSETS AND PLANAR GRAPHS STEFAN FELSNER AND WILLIAM T. TROTTER Abstract. Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [ t − 1 l t ] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [ 2 l 3 ]. This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3. • The largest n for which the dimension of the complete graph Kn is at most [ t − 1 l t ] is the number of antichains in the lattice of all subsets of a set of size t − 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind’s problem. This result extends work of Hoşten and Morris [14]. The main results are enriched by background material which links to a line of reserch in extremal graph theory which was stimulated by a problem posed by G. Agnarsson: Find the maximum number of edges in a graph on n nodes with dimension at most t. 1. Introduction Let G = (V, E) be a finite simple graph. Definition 1.1. A nonempty family R of linear orders on the vertex set V of a graph G = (V, E) is called a realizer of G provided (∗) For every edge S ∈ E and every vertex x ∈ X − S, there is some L ∈ R so that x > y in L for every y ∈ S. The dimension of G, denoted dim(G), is then defined as the least positive integer t for which G has a realizer of cardinality t. In order to avoid trivial complications when the condition (∗) is vacuous, throughout the remainder of the paper, we restrict our attention to connected graphs with three or more vertices. For those readers who are new to the concept of dimension for graphs, we present the following elementary example. Example 1.2. The dimension of the complete graph K5 is 4, but the removal of any edge reduces the dimension to 3. 1991 Mathematics Subject Classification. 06A07, 05C35. Key words and phrases. Dimension, planarity, outerplanarity. The research of the second author was supported in part by the Office of Naval Research and the Deutsche Forschungsgemeinschaft. 1 2 S. FELSNER AND W. T. TROTTER Proof. Consider the complete graph with vertex set {1, 2, 3, 4, 5}. Any family of 4 linear orders {L1 , L2 , L3 , L4 } with i the highest element and 5 the second highest element in Li for all i is a realizer. So dim(K5 ) ≤ 4. On the other hand, suppose dim(K5 ) ≤ 3, and let R = {M1 , M2 , M3 } be a realizer. Without loss of generality, 4 and 5 are not the highest element of any linear order in R. Also, without loss of generality 4 > 5 in both M1 and M2 . Now let j be the largest element of M3 . Then there is no element i ∈ {1, 2, 3} for which 5 is over both 4 and j in Mi . The contradiction shows that dim(K5 ) = 4, as claimed. Now let e = {3, 4}. The following three linear orders form a realizer of K5 − e: L1 = [2 < 3 < 5 < 4 < 1] L2 = [1 < 3 < 5 < 4 < 2] L3 = [1 < 2 < 4 < 5 < 3]  Here is a second example. We leave its elementary proof as an exercise. Example 1.3. The dimension of the complete bipartite graph K3,3 is 4, but the removal of any edge reduces the dimension to 3. The preceding two examples help to motivate the following, now classic, theorem of W. Schnyder [16]. Theorem 1.4. A graph G is planar if and only if its dimension is at most 3.  Schnyder’s original version of Theorem 1.4 used a slightly different concept of dimension. With a finite graph G = (V, E), we associate a height two poset P = PG whose ground set is V ∪ E. The order relation is defined by setting x < S in PG if x ∈ V , S ∈ E and x ∈ S. PG is called the incidence poset of G. Schnyder proved: A graph G is planar if and only if the dimension of its incidence poset is at most 3. The close relationship between the dimension of a graph and the dimension of its incidence poset can be described as follows: Proposition 1.5. Let G be a graph and let PG be its incidence poset. Then (1) dim(G) ≤ dim(PG ) ≤ 1 + dim(G). (2) dim(G) = dim(PG ) if G has no vertices of degree 1.  Although the preceding proposition admits an elementary proof, it can be stated in a somewhat more general form: Proposition 1.6. The dimension of a graph equals the interval dimension of its incidence poset.  Graphs and incidence orders of dimension at most two are easy to characterize: Proposition 1.7. Let G be a graph and let PG be its incidence poset. Then (1) dim(G) ≤ 2 if and only if G is a caterpillar. (2) dim(PG ) ≤ 2 if and only G is a path.  We will not use the concepts of dimension and interval dimension for posets extensively in this article, but for those readers who would like additional information on how this parameter relates to graph theory problems, we suggest looking at Trotter’s monograph [20] or survey articles [21], [22] and [23]. Although we do not include a proof of Schnyder’s theorem here, we pause for some comments related to it. POSETS AND PLANAR GRAPHS 3 The fact that graphs with dim(G) ≤ 3 are planar is relatively easy to prove. This was shown by Babai and Duffus [3]. The difficult part is to show that dim(G) ≤ 3 when G is planar. This proof required Schnyder to develop several elegant structural results for planar graphs, and these results have interest independent from their application to Theorem 1.4. Schnyder’s theorem has been generalized by Brightwell and Trotter [5], [6] with the following two results. Theorem 1.8. Let D be a plane drawing without edge crossings of a 3-connected planar graph G and let P be the poset of vertices, edges and faces of this drawing, partially ordered by inclusion. Then dim(P) = 4. Furthermore, the subposet of P generated by the vertices and faces is 4-irreducible.  Theorem 1.9. Let D be a plane drawing without edge crossings of a planar multigraph G and let P be the poset of vertices, edges and faces of this drawing, partially ordered by inclusion. Then dim(P) ≤ 4.  Simplified proofs of Theorem 1.8 have been given by Felsner [10], [11]. 2. Other Combinatorial Connections In order to provide further motivation for the results which follow, we pause to discuss two other recent research directions. One such theme is to determine (or estimate) the dimension of the complete graph Kn . Note that the dimension of Kn and the dimension of its incidence poset are the same when n ≥ 3. For a positive integer t, let B(t) denote the set of all subsets of {1, 2, . . . , t}. A subset A ⊂ B(t) is called an antichain if no two sets in A are ordered by inclusion. We then let D(t) count the number of antichains1 Starting with D(1) = 3 the next values are: 6, 20, 168, 7781. Exact values are known for t ≤ 8 The evaluation (or estimation) of the function D(t) is popularly known as Dedekind’s Problem [18]. We then let HM(t) count the number of antichains A in B(t) which satisfy the following additional property: (∗∗) S1 ∪ S2 6= {1, 2, . . . , t} for every S1 , S2 ∈ A. Starting with HM(1) = 2 the next values are: 4, 12, 81. These numbers arise in several combinatorial problems [18], but here is one particularly surprising one recently discovered by Hoşten and Morris [14]. Theorem 2.1. Let t ≥ 2. Then HM(t−1) is the largest n so that dim(Kn ) ≤ t.  So it is natural to ask whether there is a connection between dimension and Dedekind’s problem which avoids the technical restriction (∗∗) described above. But perhaps there is even a more significant motivation involving minor-monotone graph parameters—a subject which has attracted considerable attention in the last few years. For example, let µ(G) denote the Colin de Verdière graph invariant introduced in [8]. The parameter µ(G) is minor-monotone. Furthermore: (1) µ(G) ≤ 1 if and only if G is a path. (2) µ(G) ≤ 2 if and only if G is outerplanar. (3) µ(G) ≤ 3 if and only if G is planar. (4) µ(G) ≤ 4 if and only if G is linklessly embeddable in 3-space. 1In this count we include the empty antichain. 4 S. FELSNER AND W. T. TROTTER We refer the reader to Schrijver’s survey article [17] for an extensive discussion of the Colin de Verdière invariant. However, in view of our previous remarks, it is striking that in the list of results for this invariant, we see both a characterization of paths and of planar graphs. So it is natural to explore the concept of dimension of graphs to see if one can find a characterization of outerplanar graphs, a characterization of linklessly embeddable graphs and a natural extension to a minor-monotone parameter. We have solved the first of these three challenges. 3. A New Characterization of Outerplanar Graphs Let L and M be linear orders on a finite set X. We say that L and M are dual and write L = M d if x < y in L if and only if x > y in M for all x, y ∈ X. Reflecting on the problem of characterizing outerplanar graphs in terms of dimension, one is also faced with the problem of finding a number between 2 and 3, this object2 will be denoted [2 l 3 ]. Definition 3.1. For an integer t ≥ 2, we say that the dimension of a graph is [t−1 l t] if it has dimension greater than t − 1 yet has a realizer of the form {L1 , L2 , . . . , Lt } with Lt−1 = Ldt . As the reader will see, the following theorem is not difficult to prove. It is the statement which is a bit surprising. Theorem 3.2. A graph G is outerplanar iff it has dimension at most [2 l 3 ]. Proof. Let G be a graph and suppose that dim(G) ≤ [2 l 3 ]. We show that G is outerplanar. Choose a realizer {L1 , L2 , L3 } for G with L2 = Ld3 . Then let H be the graph formed by adding a new vertex x adjacent to all vertices of G. We show that H is planar. To accomplish this, consider the family R = {M1 , M2 , M3 } of three linear orders on the vertex set of H formed by adding x at the top of L1 , the bottom of L2 and the bottom of L3 . We claim that R is a realizer of H. To see this, let u be a vertex in H and let f be an edge not containing u as one of its endpoints. If u = x, then x is over both points of f in M1 . So we may assume u 6= x. If f = {x, v}, with v a vertex from G and u 6= v, then u is over both x and v in exactly one of M2 and M3 . Finally, if f = {v, w}, where both v and x are vertices in G, then there is some i ∈ {1, 2, 3} for which u is over both v and w in Li . It follows that u is over v and w in Mi . Thus by Schnyder’s theorem, H is planar. In turn, G is outerplanar. Now suppose that G is outerplanar. We show that the dimension of G is at most [2 l 3 ]. Without loss of generality, we may assume that G has n ≥ 4 vertices and is maximal outerplanar, i.e., adding any missing edge to G produces a graph which is no longer outerplanar. As before, let H be formed from G by adding a new vertex x adjacent to all vertices of G. Then H is maximal planar. Choose a plane drawing without edge crossings of H so that the vertex x appears on the exterior triangle. Let u1 and un be the other two vertices on this triangle. Then there is a natural labelling of the vertices of G as u1 , u2 , . . . , un so that {ui , ui+1 } is an edge and {x, ui , ui+1 } is a triangular face in the drawing for all i = 1, 2, . . . , n − 1. Let L2 be the subscript order u1 < u2 < · · · < un and let L3 be the dual of L2 . 2In the original manuscript we have used the fraction 5 for this purpose. This, however, could 2 be confused with with the independent notion of fractional dimension (see [4], [12]). POSETS AND PLANAR GRAPHS u1 u2 u5 u7 x 5 u10 Figure 1. An example for the construction. Shortest path trees for u1 and u10 are color coded. A corresponding permutation L1 is u1 , u10 , u3 , u2 , u9 , u4 , u8 , u6 , u5 , u7 , x. Call a path ui1 , ui2 , . . . , uir in G monotonic if i1 < i2 < · · · < ir . For each integer i with 1 < i < n, note that there is a unique shortest monotonic path P (u1 , ui ) from u1 to ui . Likewise, there is a unique shortest monotonic path P (ui , un ) in G from ui to un . Then let Si be the region consisting of all points in the plane belonging to the closed region bounded by the edges in these two paths together with the edge {u1 , un }. By convention, we take S1 and Sn as the degenerate region consisting of those points in the plane which are on the edge {u1 , un }. Define a strict partial order Q on the set {u1 , u2 , . . . , un } by setting ui < uj in Q if and only if Si is a proper subset of Sj . Then let L1 be any linear extension of Q. We claim that {L1 , L2 , L3 } is a realizer of G. To see this, let u be a vertex of G and let e = {y, z} be an edge not containing u. We show that there is some i ∈ {1, 2, 3} for which u is over both y and z in Li . This conclusion is straightforward except possibly when there exist integers i, j, k with 1 ≤ i < j < k ≤ n so that {y, z} = {ui , uk } and u = uj . However, in this case, it is easy to see that u is over y and z in L1 .  4. The Connection with Dedekind’s Problem In this section, we show that our refined dimension concept for complete graphs yields a full equivalence with the classical problem of Dedekind. Again, the proof is not difficult, and we find the statement the real surprise. Theorem 4.1. For t ≥ 3 the largest n so that dim(Kn ) ≤ [t − 1 l t] is D(t − 2). Proof. We first show that if dim(Kn ) ≤ [t − 1 l t], then D(t − 2) ≥ n. Let R = {L1 , L2 , . . . , Lt } be a realizer which shows that dim(Kn ) ≤ [t − 1 l t]. By relabelling, we may assume that: (1) The vertex set of Kn is {1, 2, . . . , n}, (2) 1 < 2 < · · · < n in Lt , and (3) 1 > 2 > · · · > n in Lt−1 . Now for each i, j ∈ {1, 2, . . . , n} with 1 ≤ i < j ≤ n, let S(i < j) = {α ∈ {1, 2, . . . , t − 2} : i < j in Lα }. Then for each i = 1, 2, . . . , n − 1, let Ci = {S(i < j) : i < j ≤ n}. Order the sets in each Ci by inclusion and let Ai denote the set of maximal elements of Ci . By construction, each Ai is an antichain in B(t − 2), in fact a non-empty antichain. Finally, set An = ∅. We claim that Ai 6= Aj for all 1 ≤ i < j ≤ n. In fact, we claim that there exists a set S ∈ Ai so that S * T for every T ∈ Aj . This is clearly true if j = n. But 6 S. FELSNER AND W. T. TROTTER suppose that this claim fails for some pair i, j with 1 ≤ i < j < n. Consider the set S(i < j). Then there is a set S ∈ Ai with S(i < j) ⊆ S. Suppose that there is a set T ∈ Aj so that S ⊆ T . Choose k with j < k ≤ n so that T = S(j < k). It follows that whenever α ∈ {1, 2, . . . , t − 2} and i < j in Lα , then j < k in Lα . So there is no α in {1, 2, . . . , t − 2} for which j is over both i and k. Since j is between i and k in both Lt−1 and Lt , it follows that R is not a realizer. The contradiction completes the first part of the proof. Now suppose that D(t − 2) ≥ n. We want to show that dim(Kn ) ≤ [t − 1 l t]. Here we only provide a sketch of the argument since it follows immediately from the next lemma, a result due to Hoşten and Morris. It is also presented in somewhat more compact form in Kierstead’s survey paper [15] and has its roots in Spencer’s paper [19], where the asymptotic behavior of the dimension of the complete graph is first discussed. First, let s ≥ 1 and let L = (S1 , S2 , . . . , S2s ) be a listing of all the subsets of {1, 2, . . . , s} so that i < j whenever Si ⊂ Sj , i.e., this listing is a linear extension of the inclusion ordering. Then suppose that D(s) = n and let A1 , A2 , . . . , An be the unique listing of the antichains in B(s) so that For all i < j with 1 ≤ i < j ≤ n, if k is the largest integer in {1, 2, . . . , 2s } so that Sk belongs to one of Ai and Aj but not the other, then Sk belongs to Ai . In other words, the listing of antichains is in reverse lexicographic order as determined by the listing L. The proof of the following lemma is given in [14]. Lemma 4.2. Let s ≥ 1, let L be a linear extension of the inclusion order on the subsets of {1, 2, . . . , s} and let A1 , A2 , . . . , An be the antichains of B(s) listed in reverse lexicographic order as determined by L. For each i and j with 1 ≤ i < j ≤ n, let k be the largest integer in {1, 2, . . . , 2s } so that Sk belongs to one of Ai and Aj but not the other, and set S(i < j) = Sk . Then for each α ∈ {1, 2, . . . , s}, the binary relation Lα = {(i, j) : α ∈ S(i < j)k ∪ {(j, i) : α 6∈ S(i < j)} is a total order on the antichains of B(s).  It is easy to see that the orders {L1 , L2 , . . . , Ls } together with the subscript order and its dual form a realizer of the complete graph of size n with the vertices being the antichains in B(s). With this observation, the proof is complete.  5. A New Extremal Graph Theory Problem G. Agnarsson [1] first proposed to investigate the following extremal graph theory problem. For integers n and t, find the maximum number ME(n, t) of edges in a graph on n vertices having dimension at most t. Agnarsson was motivated by ring theoretic problems which are discussed in [1] and [2]. Based on the results presented thus far, we can also attempt to find the maximum number of edges ME(n, [t − 1 l t]) in a graph on n vertices having dimension at most [t − 1 l t]. For small values, we know everything, since we are just counting respectively the maximum number of edges in a caterpillar, an outerplanar graph and a planar graph. Proposition 5.1. For n ≥ 3, ME(n, 2) = n − 1, ME(n, [2 l 3 ]) = 2n − 3 and ME(n, 3) = 3n − 6.  POSETS AND PLANAR GRAPHS 7 In [2], Agnarsson, Felsner and Trotter investigated the asymptotic behavior of ME(n, 4) and used Turán’s theorem [24], the product Ramsey theorem (see [13], for example) and the Erdős/Stone theorem [9] to obtain the following result. Theorem 5.2. lim n→∞ 3 ME(n, 4) = . n2 8  The lower bound in this formula comes from the fact that any graph with chromatic number at most 4 has dimension at most 4. So the Turán graph, a balanced complete 4-part graph has dimension at most 4. This is enough to show that limn→∞ ME(n, 4)/n2 ≥ 3/8. Theorem 5.3. lim n→∞ 1 ME(n, [3 l 4 ]) = . n2 4 Proof. As the argument is a straightforward modification of the proof of Theorem 5.2, we provide only a sketch. First, note that the balanced complete bipartite graph has dimension at most [3 l 4 ] and has ⌈n2 /4⌉ edges. This shows that 1/4 is a lower bound for the limes. Now suppose that ǫ > 0 and G is any graph on n vertices with more than (1/4 + ǫ)n2 edges. We show that dim(G) > [3 l 4 ] provided n is sufficiently large. Suppose that dim(G) ≤ [3 l 4 ] and choose a realizer R = {L1 , L2 , L3 , L4 } with L3 = Ld4 . From the Erdös/Stone theorem, we know that for every p ≥ 1, G contains a complete 3-partite graph with p vertices in each part—provided n is sufficiently large in terms of p. Choose such a subgraph and label the three parts as V1 , V2 and V3 . Using the product ramsey theorem, it follows that if p is sufficiently large, there exists W1 ⊂ V1 , W2 ⊂ V2 and W3 ⊂ V3 , with |W1 | = |W2 | = |W3 | = 2, so that for each i, j, k = 1, 2, 3 with i 6= j, either all points of Wi are under all points of Wj in Lk or all points of Wi are over all points of Wj in Lk . Label the points so that W1 = {x1 , x2 }, W2 = {y1 , y2 } and W3 = {z1 , z2 }. Without loss of generality, we may assume that x1 < x2 < y1 < y2 < z1 < z2 in L3 , so that z2 < z1 < y2 < y1 < x2 < x1 in L4 . Consider the vertex y1 and the edge {x1 , y2 }. Since y1 < y2 in L3 and y1 < x1 in L4 , we may assume without loss of generality that y1 is over both x1 and y2 in L1 . Thus y1 and y2 are over x1 and x2 in L1 . Similarly, considering the vertex y2 and the edge {z1 , y1 }, we may conclude that y2 is over both z1 and y1 in L2 . Thus y1 and y2 are over z1 and z2 in L2 . Following this pattern, we may then conclude that z1 is over both z2 and y1 in L1 , while x2 is over both x1 and y1 in L2 . It follows that the middle two points of W1 ∪ W2 ∪ W3 in each of the four linear orders are y1 and y2 . This is a contradiction, since it implies that y1 is never higher than both x1 and z1 . The contradiction completes the proof.  Remark. A previous version of this paper contained two conjectures regarding the structure of the extremal graphs of dimension at most [3 l 4 ] and 4. The conjectures where that these graphs can be obtained from complete four-partite and bipartite graphs by adding an maximal outerplaner graph on each of the color classes. 8 S. FELSNER AND W. T. TROTTER Both of these conjectures have been disproved recently by de Mendez and Rosenstiehl [7]. An independent example disproving the second of the conjectures was brought to our attention by an anonymous referee. 6. Minor-monotone Issues It follows from Schnyder’s theorem that the property of having dimension at most 3 is minor closed, i.e., if G has dimension at most 3, then any minor of G has dimension at most 3. However, we no of no direct proof of this assertion—other than to appeal to the full power of Schnyder’s theorem. Ideally, one would like to find an alternative proof of Schnyder’s theorem by combining the following three assertions: (1) For every n ≥ 1, the n × n grid has dimension at most 3. (2) If G is a planar graph, there is some n ≥ 1 for which G is a minor of an n × n grid. (3) Every minor of a graph of dimension at most 3 has dimension at most 3. Of course, each of these three statements is true, and simple proofs are known for the first two. So we just want to find a direct proof of the third. We also know that the property of having dimension at most [2 l 3 ] is minor closed. However, we do not know of a simple proof of this statement either. For t ≥ [3 l 4 ], it is easy to see that the property dim(G) ≤ t is no longer minor closed. For example, dim(Kn ) → ∞ but if we subdivide each edge, then we obtain a bipartite graph which has dimension at most [3 l 4 ]. We may then ask whether there is an appropriate generalization of the concept of dimension which coincides with the original definition when t < [3 l 4 ] and is minor closed when t ≥ [3 l 4 ]. We could also ask whether there is any way to characterize linklessly embeddable graphs in this framework. 7. Complexity Issues Yannakakis [25] showed that testing for dim(P) ≤ t is NP-complete for every fixed t ≥ 3. Yannakakis also proved that testing for dim(P) ≤ t is NP-complete even for height 2 posets when t ≥ 4. However, he was not able to settle whether testing for dim(P) ≤ 3 is NP-complete for height 2 posets. This problem remains open. Our original definition for dimension was formulated for a graph. However, it applies equally as well to hypergraphs. In a similar manner, we can speak of the incidence poset PH of a hypergraph H. When G is a graph, testing for dim(G) ≤ 3 is linear, since this is just a test for planarity. A similar remark holds when testing for dim(G) ≤ [2 l 3 ]. When H is a hypergraph, we do not know if testing for dim(H) ≤ 3 is NP-complete. Also, we do not know whether testing for dim(H) ≤ [2 l 3 ] is NP-complete. We suspect that testing for dim(G) ≤ [3 l 4 ] is NP-complete, but have not been able to settle the question. Acknowledgement. The authors would like to thank Walter D. Morris, Jr., Serkan Hoşten and Geir Agnarsson for sharing preliminary versions of their papers with us. We would also like to thank them for numerous electronic communications, all of which were valuable to our investigations. POSETS AND PLANAR GRAPHS 9 References [1] G. 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Turán, On an extremal problem in graph theory (in Hungarian), Matematikai és Fizikai Lapok 48 (1941), 436–452. [25] M. Yannakakis, On the complexity of the partial order dimension problem, SIAM J. Alg. Discr. Meth. 3 (1982), 351–358. Technische Universität Berlin, Institut für Mathematik, MA 6-1 Strasse des 17. Juni 136, 10623 Berlin, Germany E-mail address: [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 303320160, U. S. A. E-mail address: [email protected]