CATERPILLAR TOLERANCE REPRESENTATIONS OF CYCLES

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1 CATERPILLAR TOLERANCE REPRESENTATIONS OF CYCLES NANCY EATON AND GLENN FAUBERT Abstract. A caterpillar, H, is a tree containing a path, P, such that every vertex of H is either in P or adjacent to P. Given a graph G a caterpillar tolerance representation of tolerance t on a caterpillar, H, is a mapping of each vertex, v V (G) to a subtree, H v H, such that uv E(G) if and only if H u and H v share at least t vertices. Given any positive integers, h and t, it is determined for all values of n whether or not a caterpillar tolerance representations exists for a cycle of length n using a representing caterpillar of maximum degree h and tolerance t. 1. Introduction We begin with a few definitions: A caterpillar is a tree which contains a path, P, such that every vertex is either in or adjacent to P. A caterpillar tolerance representation of a graph G is a function φ : V (G) S where S is the family of subtrees of a fixed caterpillar, H, called the host. Each vertex, v, of G is assigned a subtree, H v, in the host caterpillar, and v i v j E(G) if and only if V (Hvi ) V (H vj ) t, where t is a positive integer called the tolerance of the representation. The set cat[h, t] will denote the set of all graphs for which a caterpillar representation of tolerance t exists in a host of maximum degree h. More generally, a tree tolerance representation is a function φ : V (G) H where H is the family of subtrees of a fixed tree, H, called the host. Each vertex, v, of G is assigned a subtree, H v, in the host tree, and v i v j E(G) if and only if V (Hvi ) V (H vj ) t, where t is a positive integer called the tolerance of the representation. The set of all graphs with a tolerance t representation by a tree of maximum degree h with subtrees of maximum degree s is denoted [h, s, t]. It has been shown by Jameson and Mulder in [4] that all graphs have a tree tolerance representation of tolerance 2. That the set of graphs with representations of tolerance 1 are precisely the chordal graphs is due, separately, to Buneman, Gavril, and Walter, and can be found in [1, 3, 6] respectively. Date: September 21,

2 2 NANCY EATON AND GLENN FAUBERT Representations for which the host is a path are well understood. The set of graphs representable with a path host is independent of tolerance, that is, [2, 2, t] = [2, 2, 1] for all positive integers, t as shown in [4]. The characterization of all graphs representable by a path involves a structure called an asteroidal triple. An asteroidal triple is a set of three vertices v 1, v 2, v 3 in the graph such that for any permutation, (v i, v j, v k ), of these vertices, there exists a path from v i to v j not adjacent to v k. In [5], Lekkerkerker and Boland have shown in the equivalent context of interval graphs, that a graph, G, can be represented by a path if and only if G is chordal and contains no asteroidal triples. Representations for which the host is a star are related to a another type of graph representation called a set representation. A p- intersection representation is a special type of set representation. Let P(X) be the power set of some base set X = {1, 2, 3,..., t}, then a p-intersection representation is a function φ : V (G) P in which v X v such that v i v j E(G) if and only if X vi X vj p. Denote by θ p (G) the least t = X for which a p-intersection representation exists for the graph G. It has been shown that for all graphs, G, and all positive integers, p, a p-intersection representation for G exists. It has also been shown that for any graph, G, a 1-intersection representation on the set of labels {1, 2, 3,..., t} exists if and only if a tree tolerance representation of tolerance 2 exists using a host star of maximum degree t. To see this, name the t degree 1 nodes of a host star with the integers 1,2,3,...,t. Each vertex, v, of G is assigned a subtree in the tree tolerance representation consisting of the central node and the same subset of the degree 1 nodes of the star as are named in the 1- intersection representation for v. From this we see that two vertices, u, v of G share k + 1 nodes of the host star if and only if the sets X u and X v intersect in exactly k elements from X. Both types of representations for any graph G can be constructed by numbering the edges of G with {1, 2, 3,..., E(G)}. For the set representation, we put x X v precisely when v is incident to an edge numbered x. The caterpillar is a generalization of the path and the star. As a generalization of the path, it provides an opportunity to explore the consequences of adding leaves to the path (i.e. feet on the spine of the caterpillar) and seeing which new graphs can be represented. As a generalization of the star, we can explore the consequences of limiting the maximum degree while joining several stars together along a spine. In this paper we explore the representability of cycles using caterpillars. The ability of a host to represent induced cycles was important in

3 CATERPILLAR TOLERANCE REPRESENTATIONS OF CYCLES 3 the work on path representations graphs and has provided a starting point for the study of caterpillar representations. To reduce repetition in proofs we will assume the following unless otherwise noted. When we say G cat[h, t] we mean that there exists a caterpillar tolerance representation, H = {H v } v V (G), of G with maximum host degree, h, and tolerance, t. For clarity the word node will be used to refer to a vertex of the host caterpillar to distinguish it from the vertices of a represented graph. Let the spine, S, of the caterpillar be a the longest path to which all other nodes of the caterpillar are adjacent. We will denote by S v the set V (H v ) V (S). Any node that is not on the spine is called a foot. Every foot is adjacent to exactly one spinal node. Because of the natural structure of cycles it will sometimes be convenient to use the notation v 0, v 1, v 2,..., v n 1 and H v0, H v1, H v2,..., H vn 1, to represent the vertices and subtrees of the cycle C n where v i v i+1 E(C n ), 0 i n 1. Assume the addition on the subscripts is modulo n. For convenience, we will often denote the spinal nodes of H as s 1, s 2,..., s m. In the host, we will say that a spinal vertex, s i, is to the left of a spinal vertex, s j, if i < j, and will in this case say, s i < s j. Similarly, we will say s i is to the right of s j just when i > j, and say s i > s j. 2. Some monotonic class membership relations A simple fact regarding caterpillar representations is that if G cat[h, t], then G cat[h, t] for all h h. Also, and more interestingly, if G cat[h, t], then G cat[h, t ] for all t t. We now will prove these monotonic class membership relations. First, the theorem about increasing maximum degree. Theorem 2.1. For h, t Z +, cat[h, t] cat[h + 1, t]. Proof. Let H be a host caterpillar of max degree h for which a representation exits. A foot can be added to any or all of the spinal nodes of H increasing the host s maximum degree by one without affecting any of the subtrees H i in the representation. Now we show monotonicity of inclusion with tolerance. Theorem 2.2. For h 2 and t 1, cat[h, t] cat[h, t + 1]. Proof. Let G cat[h, t] with host H and representation H. We construct a new host tree, H and a new representation H as follows: To the old host, H having m spinal nodes, we appending a new leaf, s m+1,

4 4 NANCY EATON AND GLENN FAUBERT adjacent to s m to form the host H. To construct the new representation, first we consider those vertices, v, of G, whose representative subtree, H v in H is a single foot f j. This foot is attached to exactly one node, s j. We set H v = H v + s j. Now we consider any of the remaining vertices, v, of G, whose representative tree, H v, will have a highest indexed spinal node, s i. We set H v = H v + s i+1. We claim that H is a maximum degree h, tolerance t + 1 caterpillar representation of G and hence that G cat[h, t+1]. To prove the claim let u and v be two distinct vertices of G. Assume first that S u S v is non empty and contains a spinal node. Let i be the largest index such that s i S u S v. Then V (H u) V (H v) = (V (H u ) V (H v )) {s i+1 }. Next assume that V (H u ) V (H v ) contains only a single foot f then at least one of H u and H v is the foot f and hence V (H u) V (H v) will contain f together with that spinal node to which f is adjacent. Note that this exhausts the ways that the trees H u and H v can have nonempty intersection. Hence if H u and H v have k nodes in common for k 1 then H u and H v must have k + 1 nodes in common. Now we may suppose that V (H u ) V (H v ) is empty. If H u and H v are feet of different spinal nodes then V (H u) V (H v) will also be empty. If H u and H v are distinct feet of the same spinal nodes, s, then V (H u) V (H v) will contain precisely one node, s. If exactly one of H u and H v is a foot of a spinal node s then V (H u) V (H v) will contain at most s. Finally, if neither of H u and H v are mere feet then assume without loss of generality that all the spinal indices of S u are less than all the spinal indices of S v. Let s i be the highest indexed spinal node of S u then V (H u) V (H v) will contain at most one node, s i Caterpillar Representations of Cycles Given n 3 we will determine the precise lower boundaries of both h and t for which C n cat[h, t]. That upper bounds do not exist for inclusion has been shown in Theorems 2.1 and 2.2 above. In each case, we will give an actual construction of a cat[h, t] representation for the minimum values of h and t for which a representation exists, and a nonexistence proof for all smaller values of h and t. The following is a simple construction that we can use to build representations of various cycles. It also shows that cycles of arbitrary length can be represented with caterpillars. Theorem 3.1. If n (h 2)(t 1) + 2 with h 3 and t 2, then C n cat[h, t]. Proof. Construct the representation as follows: Choose t 1 consecutive nodes along the spine of an h-regular caterpillar and assign every one

5 CATERPILLAR TOLERANCE REPRESENTATIONS OF CYCLES 5 of them to each vertex of C n. The neighborhood of this set of t 1 nodes consists of (h 2)(t 1) feet plus two more spinal nodes. Each one of these (h 2)(t 1) + 2 nodes is assigned to a different pair of adjacent vertices in C n. Adjacent vertices of C n will share t nodes while non-adjacent pairs will only share t 1 nodes. The following corollary of Theorem 3.1, shows us that we can represent any length cycle using only caterpillars of degree three if we are willing to use a high enough tolerance. Corollary 3.2. The cycle C n cat[3, n 1]. Proof. The proof follows directly from Theorem 3.1, We will now break our investigation into cases according to the tolerance of the representation. There are only four cases. All caterpillar tolerance representations for cycles with tolerances of 4 or greater can be handled together. In every case we will assume that h 2. Caterpillars of maximum degree zero or one are trivial Cycles in caterpillar representations of tolerance 1. It is easy to see that C 3 cat[h, 1] for any value of h. Simply assign every vertex of C 3 to the same node of the host. As mentioned earlier, C 4 / cat[h, 1] for any h since C 4 is not chordal. Hence we can dispense with the case where t=1 by noting that for h 2, C 3 cat[h, 1] and C 4 / cat[h, 1] Cycles in caterpillar representations of tolerance 2. In this section we will see that C n cat[h, 2] n h. The following corollary follows from Theorem 3.1 when t = 2. We will finish this case by showing that C n / cat[n 1, 2]. Corollary 3.3. For n 3, C n cat[n, 2]. Proof. The proof follows directly from Theorem 3.1, n (n 2)(2 1) + 2 for n 3, hence C n cat[n, 2]. Theorem 3.4. If n 4, then C n / cat[n 1, 2]. Proof. If such a representation exists, each subtree, H v, must include at least three nodes from host H. Indeed, since for n 4, all vertices of C n have a pair of non-adjacent neighbors, each subtree must include at least three nodes. Moreover, each subtree must contain a node from the spine of H since no two host feet are adjacent. First we show that the representative subtrees of any two vertices can share at most one spinal node of H. Clearly, the subtrees of any two

6 6 NANCY EATON AND GLENN FAUBERT nonadjacent vertices of G must share at most one spinal node. Suppose H vi and H vi+1 share two spinal nodes. Let s x be the leftmost spinal node in S vi S vi+1 and let s y be the rightmost spinal node in S vi S vi+1. Let P xy be the unique path along the spine of H with end points s x and s y. Without loss of generality, we may assume that s x is strictly left of y, that the spinal node to the right of s y is not in H vi, and the spinal node to the left of s x is not in H vi+1. Let e be an edge of P xy. For j i, i + 1, it is clear that e S vj. This implies that for all j i, i + 1, S vj {s y, s y+1,..., s m } = R or S vj {s 1,..., s x 1, s x } = L. But then since v i+2, v i+3,..., v i 1 is a path in C n, then either i 1 R or i 1 L. If i 1 R we would have that S vi 1 S vi = {s y } and hence H vi 1 H vi = s y, f y. And since H vi 1 contains at least 3 nodes, s y+1 S vi 1 implying s y+1 / S vi+1 since s y H vi 1 H vi+1. But then since H vi+1 contains at least 3 nodes, H vi+1 = H vi+1 = s x, s y, f y. We reach a similar contradiction when we consider i 1 L. Thus we can conclude that s x = s y and it follows that no pair of subtrees can share two spinal nodes. We may now assume that all pairs of representative subtrees share at most one spinal node. Let v i 1, v i, v i+1 be three distinct, consecutive vertices of C n, n 4, and suppose s x and s y are spinal nodes of H vi, with s x H vi and s y H vi+1. Assume without loss of generality that s x < s y, and let P xy be the unique path in H from s x to s y. Let e be an edge of P x,y. For j i it is clear that e S vj. This implies that for all j i, S vj {s y, s y+1,..., s m } = R or S vj {s 1,..., s x 1, s x } = L. But then since v i+2, v i+3,..., v i 2 is a path in C n, either i 2 R or i 2 L. If i 1 R then S vi 2 S vi 1 = which contradicts that v i 2 and v i 1 are adjacent. We get a similar contradiction when we consider i 2 L. Thus the subtrees of any three consecutive vertices of C n share exactly one spinal node, and hence all n of the representative subtrees hold in common exactly one spinal node, say w. Since each subtree contains the spinal node w, to avoid triangles it is required that each pair of adjacent vertices on C n shares a unique neighbor of w. Since w has n adjacent vertex pairs competing for only n 1 neighbors, C n cannot be represented Cycles in caterpillar representations of tolerance 3. From Theorem 3.1 we get that C 4 cat[3,3]. It is an interesting border case because, as we will see, if t 3 and if C 5 cat[h, t] then C n cat[h, t] for all n 5.

7 CATERPILLAR TOLERANCE REPRESENTATIONS OF CYCLES 7 We will be needing new constructions now that we are discussing tolerance 3 representations. Theorem 3.1 will not get us very far. The best Theorem 3.1 can do in the tolerance 3 case is provide that C n cat[ n+2,3]. But we can say more. The following theorem gives us a 2 new construction that turns out to be versatile. Theorem 3.5. For all n, C n cat[4, 3]. Proof. By construction. We set the subtree H vi for 0 i n 2 to consist of the two spinal nodes s i and s i+1 together with the four feet attached to those spinal nodes. H vn consists of all spinal nodes s 1 through s n plus the two feet of s 1 and the two feet of s n. Our new found construction might encourage us to look for a new way to represent cycles in cat[3,3] so we might get past the limit of C 4 obtained by Theorem 3.1. Such hope is dashed, by the following theorem. The fact that C 4 is the longest cycle in cat[3,3], and that all cycles can be found in cat[4,3] shows there is something very different about tolerance 3 than we have seen in tolerances 1 and 2. Theorem 3.6. For n 5, C n / cat[3, 3]. Proof. We assume that a cat[3,3] representation exists for some n 5. Notice that in any cat[3,3] representation of any cycle of length greater than three, each representative subtree must have at least four nodes to avoid chords, and each must use at least two spinal nodes. Also, the subtrees of any two adjacent vertices must share at least two spinal nodes. First we claim that no representative subtree in such a representation can use exactly two nodes of the spine. Indeed, let v 1 be a vertex of C n which uses only two spinal nodes, s i and s i+1, of the host H in its representation. It follows that H v1 must also include the two foot nodes, f i and f i+1. The subtree H v2 must include s i and s i+1 as well as exactly one of the feet, f i or f i+1. The subtree H vn must include s i and s i+1 together with that foot of H v1 not in H v2. The particular choice of feet does not affect the argument. Without loss of generality we can assume that all spinal nodes of H v2 extend only to the right and those of H v0 extend only to the left. They must extend in different directions because v 0 and v 2 are not adjacent. Moreover, H v0 must include s i 1 else H v0 will coincide with H v1 producing the unwanted chord v 0 v 2, and H v2 must include s i+2 else H v2 will coincide with H v1 producing the same unwanted chord v 0 v 2. For j 1, {s i, s i+1 } = S vj since then H vj = S v1. We have {s i, s i+1, s i+2 } S v2 so if {s i, s i+1, s i+2 } S vj then j = 3. We also have {s i 1, s i, s i+1 } S v0 so if {s i 1, s i, s i+1 }

8 8 NANCY EATON AND GLENN FAUBERT S vj then j = n 1. We know that s i 1 / S v3 and s i+2 / S vn 1. Therefore, for j n 1, 0, 1, 2, 3, S vj {s i+1, s i+2,..., s m } = R or S vj {s 1,..., s i 1, s i } = L. If n = 5 then S v4 S v3 = {s i, s i+1 } and since H v4 and H v3 must share one other node, it must be a foot from {f i, f i+1 }, but then both v 4 and v 3 would be adjacent to v 1. If n > 5, since v 4,..., v n 2 is a path,either n 2 k=4 S v k R or n 2 k=4 S v k L. If n 2 k=4 S v k R then Svn 2 S vn 1 1 thus V (Hvn 2 ) V (H vn 1 ) 2. We reach a similar contradiction when we consider n 2 k=4 S v k R. Hence we conclude that no representative subtree in any cat[3,3] representation of C n, n 5, can use exactly two nodes of the spine. Next we claim that no two subtrees, H vj, H vj+1 can intersect in three or more spinal nodes. (Plainly, subtrees of nonadjacent vertices could not intersect in three or more spinal nodes.) Without loss of generality we may assume s i 1 / S v1 and s i+r+1 / S v0. Assume S v1 S v0 contains s i, s i+1,..., s r, r 2. Let P be the path in H from s i to s i+r. For j 0, 1, P S vj. In fact, at most 2 vertices from P are in S vi. So either S vj {s i+r 1, s i+r,..., s m } = R or S vj {s 1,..., s i, s i+1 } = L. If S vr R and S vs L then v r cannot be adjacent to v s in C n. So, since v 2, v 3,..., v n 1 is a path in C n, n 1 k=2 S v k L or n 1 k=2 S v k R. If n 1 k=2 S v k L, then for H v2 and H v1 to have at least 3 vertices in common, s i, s i+1 S v2. In fact, since S v2 3, {s i 1, s i, s i+1 } S v2. But then S v0 cannot contain s i 1. This implies that S vn 1 S v0 = {s i, s i+1 } and since Svn 1 3, {si 1, s i, s i+1 } S vn 1. But since n 5, n 1 4 and v 4 is not adjacent to v 2. We reach a similar contradiction when considering n 1 k=2 S v k R. Thus our assumption that some two subtrees of our construction can intersect in three or more nodes is false. Hence, each pair of subtrees corresponding to adjacent vertices of C n must contain exactly two nodes. Now we can assume that each subtree uses at least three spinal nodes and that no two subtrees intersect along the spine in more than two spinal nodes. Let H v1 and H v0 intersect along the spine in spinal nodes s i and s i+1. Assume without loss of generality that s i+1 is the rightmost spinal node of H v0 and that s i is the leftmost spinal node of H v1. Because each subtree uses at least three spinal nodes and no two subtrees may intersect along the spine in more than two spinal nodes, all spinal nodes of H v2 must be to the right of s i+1 inclusive. In fact the conditions on our assumed construction force all spinal nodes of H vj to be to the right of s i+j 1, j = 1... n 1 forcing the distance in H between H vn 1 and H v0 to be at least (i+n 1 1) i which is at least 3 for n 5. Without an overlap of at least two spinal nodes between the subtrees of H vn 1 and H v0, there is no chance of representing the edge

9 CATERPILLAR TOLERANCE REPRESENTATIONS OF CYCLES 9 between v n 1 and v 0. Thus we conclude that no cat[3,3] representation exists for C n, n Cycles in caterpillar representations of tolerance 4 or more. This will be the last section we will need to characterize the representability of cycles with caterpillars. We will just need one more construction which will show that C n cat[3,4]. Then we can appeal to Theorems 2.1 and 2.2 to finish up the job. Theorem 3.7. For all n, C n cat[3, 4]. Proof. By construction. The subtree H vi for 1 i n 1 consists of spinal nodes s i, s i+1 and s i+2 together with the three feet attached to the spinal nodes. H vn consists of all spinal nodes s 1 through s n+1 plus the lone foot of s 1 and the lone foot of s n A summary of Results and Further Questions The following table summarizes results for cat[h, t] representations of cycles with h 2 and t 1. Note that the maximum cycle size is indicated and all smaller cycles are included. The symbol * is used to show values determined by previous results, see [1],[3],[5],[6]. t = 1 t = 2 t = 3 t 4 h = 2 C 3 C 3 C 3 C 3 h = 3 C 3 C 3 C 4 all h = 4 C 3 C 4 all all h 5 C 3 C h all all Table 1 showing the maximum cycle representable by cat[h, t]. A look at the results for caterpillar representations of cycles summarized in table 1, suggests the possibility that some of the classes cat[h, t] may be the same. And clearly, some are different. The authors have shown cat[3,1] = cat[3,2], cat[3,1] = cat[h,1] for h > 3, and cat[4,2]=cat[3,3] in [2]. References [1] P. Buneman. A characterization of rigid cicuit graphs, Discrete Math. 9 (1974), [2] N. Eaton, G. Faubert. Caterpillar Tolerance Representations, Manuscript. [3] F. Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combinatorial Theory Ser. B 16 (1974),

10 10 NANCY EATON AND GLENN FAUBERT [4] R. E. Jameson and H. M. Mulder. Constant tolerance representations of graphs in trees, Proceedings of the Thirtieth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL 2000) Congr. Numer., vol. 143, 2000, [5] C. G. Lekkerkerker and J. Ch. Boland. Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962/1963), [6] J. R. Walter. Representations of chordal graphs as subtrees of a tree, J. Graph Theory 2 (1978) no. 3, University of Rhode Island, Kingston, RI address: eaton@math.uri.edu University of Rhode Island, Kingston, RI address: gfaubert@math.uri.edu