A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs with a vital linkage of order 2: they are certain minors of a family of highly structured graphs

Mayhew, D.

Whittle, G. (Geoff)

Zwam, S.H.M. (Stefan) van

English

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The structure of graphs with a vital linkage oforder 2∗Dillon Mayhew† Geoff Whittle† Stefan H. M. van Zwam‡§June 28, 2010AbstractA linkage of order k of a graph G is a subgraph with k components,each of which is a path. A linkage is vital if it spans all vertices, and noother linkage connects the same pairs of end vertices. We give a charac-terization of the graphs with a vital linkage of order 2: they are certainminors of a family of highly structured graphs.1 IntroductionRobertson and Seymour [4] defined a linkage in a graph G as a subgraphin which each component is a path. The order of a linkage is the numberof components. A linkage L of order k is unique if no other collection ofpaths connects the same pairs of vertices, it is spanning if V (L) = V (G),and it is vital if it is both unique and spanning. Graphs with a vital link-age are well-behaved. For instance, Robertson and Seymour proved thefollowing:Theorem 1.1 (Robertson and Seymour [4, Theorem 1.1]). There exists aninteger w, depending only on k, such that every graph with a vital linkageof order k has tree width at most w.Note that Robertson and Seymour use the term p-linkage to denote alinkage with p terminals. Robertson and Seymour’s proof of this theoremis surprisingly elaborate, and uses their structural description of graphswith no large clique-minor. Recently Kawarabayashi and Wollan [2] gavea shorter proof that avoids using the structure theorem.∗The research of all authors was partially supported by a grant from the Marsden Fund ofNew Zealand. The first author was also supported by a FRST Science & Technology post-doctoralfellowship. The third author was also supported by NWO, grant 613.000.561.†School of Mathematics, Statistics and Operations Research, Victoria University of Welling-ton, New Zealand. E-mail: Dillon.Mayhew@msor.vuw.ac.nz, Geoff.Whittle@msor.vuw.ac.nz‡University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1§Centrum Wiskunde en Informatica, Postbus 94079, 1090 GB Amsterdam, The Netherlands.E-mail: Stefan.van.Zwam@cwi.nl1Figure 1: The graph XX.Figure 2: The graphs U¨4 and U¨5.Our interest in linkages, in particular those of order 2, stems fromquite a different area of research: matroid theory. Truemper [5] studieda class of binary matroids that he calls almost regular. His proofs leanheavily on a class of matroids that are single-element extensions of thecycle matroids of graphs with a vital linkage of order 2. These matroidsturned up again in the excluded-minor characterization of matroids thatare either binary or ternary, by Mayhew et al. [3].Truemper proves that an almost regular matroid can be built from oneof two specific matroids by certain∆−Y operations. This is a deep result,but it does not yield bounds on the branch width of these matroids. In aforthcoming paper the authors of this paper, together with Chun, will givean explicit structural description of the class of almost regular matroids[1]. The main result of this paper will be of use in that project.To state our main result we need a few more definitions. Fix a graphG and a spanning linkage L of order k. A path edge is a member of E(L);edges in E(G) − E(L) are called rung edges. A linkage minor of G withrespect to L is a minor H of G such that all path edges in E(G)−E(H) havebeen contracted, and all rung edges in E(G)− E(H) have been deleted. Ifthe linkage L is clear from the context we simply say that H is a linkageminor of G.The graph XX is depicted in Figure 1. For each integer n, the graph U¨nis the graph with V (U¨n) = {v1, . . . , vn} ∪ {u1, . . . , un}, andE(U¨n) ={vi vi+1 | i = 1, . . . , n− 1} ∪ {uiui+1 | i = 1, . . . , n− 1}∪{ui vi | i = 1, . . . , n} ∪ {ui vn+1−i | i = 1, . . . , n}. (1)We denote by Ln the linkage of U¨n consisting of all edges vi vi+1 and uiui+1for i = 1, . . . , n− 1. In Figure 2 the graphs U¨4 and U¨5 are depicted.Finally, we say that G is a Truemper graph if G is a linkage minor of U¨nfor some n. The main result of this paper is the following:Theorem 1.2. Let G be a graph. The following statements are equivalent:(i) G has a vital linkage of order 2;(ii) G has a spanning linkage of order 2 with no XX linkage minor;(iii) G is a Truemper graph.2Figure 3: The graph U¨6. The linkage is formed by the two diagonally drawnpaths.Robertson and Seymour [4] commented, without proof, that graphswith a vital linkage with k ≤ 5 terminal vertices have path width at mostk. A weaker claim is the following:Corollary 1.3. Let G be a graph with a vital linkage of order 2. Then G haspath width at most 4.Another consequence of our result is that graphs with vital linkage oforder 2 are projective-planar:Corollary 1.4. Let G be a graph with a vital linkage of order 2. Then G canbe embedded on a Möbius strip.Both corollaries can be seen to be true by considering an alternativedepiction of U¨2n, analogous to Figure 3.2 Proof of Theorem 1.2We start with a few more definitions. Suppose L is a linkage of order 2with components P1 and P2, such that the terminal vertices of P1 are s1and t1, and those of P2 are s2 and t2. We order the vertices on the pathsin a natural way, as follows. If v and w are vertices of Pi , then we saythat v is (strictly) to the left of w if the graph distance from si to v in thesubgraph Pi is (strictly) smaller than the graph distance from si to w. Thenotion to the right is defined analogously.We will frequently use the following elementary observation, whoseproof we omit.Lemma 2.1. Let G be a graph with spanning linkage L of order 2. Let P1and P2 be the components of L, with terminal vertices respectively s1, t1 ands2, t2. Let H be a linkage minor of G. If v and w are on Pi , and v is to theleft of w, then the vertex corresponding to v in H is to the left of the vertexcorresponding to w in H.Without further ado we dive into the proof, which will consist of asequence of lemmas. The first deals with the equivalence of the first twostatements in the theorem.Lemma 2.2. Let G be a graph with a spanning linkage L of order 2. ThenL is vital if and only if G has no XX linkage minor with respect to L.3s1 t1s2 t2v1v2v′2v′1u′1u′2u2u1Figure 4: Detail of the proof of Lemma 2.2.s1 t1s2 t2e fe′ f′Figure 5: Detail of the proof of Lemma 2.2.Proof. First we suppose that there exists a graph G with a non-vital span-ning linkage L of order 2 such that G has no XX linkage minor. Let P1,P2 be the paths of L, where P1 runs from s1 to t1, and P2 runs from s2to t2. Let P′1, P′2 be different paths connecting the same pairs of vertices.Let e = v1v2 be an edge of P ′1 such that the subpath s1 − v1 of P ′1 is alsoa subpath of P1, but e is not an edge of P1. Let f = u2u1 be an edge ofP ′1 such that the subpath u1 − t1 of P ′1 is also a subpath of P2, but f isnot an edge of P2. Similarly, let e′ = v′2v′1 be an edge of P ′2 such that thesubpath s2 − v′2 of P ′2 is also a subpath of P2, but e′ is not an edge of P2.Let f ′ = u′1u′2 be an edge of P ′2 such that the subpath u′2 − t2 of P ′2 is alsoa subpath of P2, but f′ is not on P2. See Figure 4.Since P ′1 and P ′2 are vertex-disjoint, v′2 must be strictly to the left ofv2 and u2. For the same reason, v′1 must be strictly between v1 and u1.Likewise, u′2 must be strictly to the right of v2 and u2, and u′1 must bestrictly between v1 and u1. Now construct a linkage minor H of G, asfollows. Contract all edges on the subpaths s1 − v1, v′1 − u′1, and u1 − t1of P1, contract all edges on the subpaths s2 − v′2, v2 − u2, and u′2 − t2 ofP2, delete all rung edges but {e, f , e′, f ′}, and contract all but one of theedges of each series class in the resulting graph. Clearly H is isomorphicto XX, a contradiction.Conversely, suppose that G has an XX linkage minor H, but that Lis unique. Let e = v1v2, f = u2u1, e′ = v′2v′1, and f ′ = u′1u′2 be the rungedges of G used by H, so that the edges in H are as in Figure 5. By Lemma2.1, v1 is strictly to the left of v′1 and u′1, and u1 is strictly to the right ofv′1 and u′1. Likewise, v′2 is strictly to the left of v2 and u2, and u′2 is strictlyto the right of v2 and u2. Now let P′1 be the union of the paths s1 − v1,v1 − v2, v2 − u2, u2 − u1, and u1 − t1. Let P ′2 be the union of the pathss2− v′2, v′2− v′1, v′1−u′1, u′1−u′2, and u′2− t2. Then L := P ′1∪ P ′2 is a linkagedistinct from L connecting the same pairs of vertices, a contradiction.Next we show that the third statement of Theorem 1.2 implies thesecond.Lemma 2.3. For all n, U¨n has no XX linkage minor with respect to Ln.4Proof. The result holds for n ≤ 2, because then |V (U¨n)| < |V (XX)|. Sup-pose the lemma fails for some n ≥ 3, but is valid for all smaller n. Everyedge of XX is incident with exactly one of the four end vertices of thepaths. Hence all rung edges incident with at least two of the four endvertices are not in any XX linkage minor. But after deleting those edgesfrom U¨n the end vertices have degree one, and hence the edges incidentwith them will not be in any XX linkage minor. Contracting these fouredges produces U¨n−2, a contradiction.For the final implication we need some more definitions. Let e = v1v2and f = u1u2 be two rung edges, with v1, u1 on P1 and v2, u2 on P2. Wesay that e and f cross if v1 is strictly to the left of u1 and v2 is strictly to theright of u2, or vice versa. We say that e and f meet if v1 = u1 or v2 = u2.A partition (A, B) of the rung edges is valid if the edges in A pairwise crossor meet, and no pair of edges in B is crossing.Reversing a path Pi means exchanging the labels of vertices si and t i ,thereby reversing the order on the vertices of the path. The followingobservation is straightforward; we omit the proof.Lemma 2.4. Let G be a graph, and L a spanning linkage of G of order 2,consisting of paths P1, running from s1 to t1, and P2, running from s2 to t2.Let G′ be obtained from G by reversing P2. If (A, B) is a valid partition ofthe rung edges of G, then (B, A) is a valid partition of the rung edges of G′.Lemma 2.5. Let G be a graph, and L a spanning linkage of order 2 of Gconsisting of paths P1, running from s1 to t1, and P2, running from s2 tot2. If G has no XX linkage minor, then there is a valid partition of the rungedges.Proof. Suppose the lemma fails for G. Choose G with as few edges aspossible. The result holds if either P1 or P2 consists of a single vertex, sothis is not the case. Let e = s1v be the leftmost edge of P1. In G/e we canpartition the rung edges into sets A and B with the desired properties. Aconflict is a pair of edges e1, e2 such that either e1, e2 ∈ A and e1, e2 do notcross and do not meet in G, or e1, e2 ∈ B and e1, e2 cross in G.Since the rung edges of G cannot be partitioned, there must be a con-flict. We assume that the valid partition (A, B) of G/e has been chosen sothat the number of conflicts in G is minimal. By reversing P2 and exchang-ing A and B as necessary, we may assume that there are edges e1, e2 ∈ Bthat cross in G. One of these, say e1, must be incident with s1, and thene2 must be incident with v.We will now refine our partition, making sure we don’t increase thenumber of conflicts in G. Suppose there is an edge f ∈ A that does notmeet or cross e1. Suppose, next, that there is an edge f′ ∈ B that crossesf . Since edges in B do not cross in G/e, f ′ must meet P2 to the right ofe1. But then G has an XX minor, a contradiction. Hence f does not crossany edge of B. It follows that the partition (A− { f }, B ∪ { f }) is a validpartition in G/e, and is moreover a partition with no more conflicts in Gthan (A, B) had.It follows that we may assume without loss of generality that everyedge f ∈ A meets or crosses e1. But then the partition (A∪ {e1}, B−{e1})5is a valid partition of G/e that has strictly fewer conflicts in G than (A, B)had, a contradiction.Lemma 2.6. Let G be a graph, and L a spanning linkage of order 2 of Gconsisting of paths P1, running from s1 to t1, and P2, running from s2 tot2. Let (A, B) be a valid partition of the rung edges. If G has no XX linkageminor, then G is a linkage minor of U¨n with respect to Ln for some integern.The lemma can be strengthened to require that U¨n has a valid partition(A′, B′) with A⊆ A′ and B ⊆ B′. We omit the proof of this fact.Proof. Suppose the statement is false. Let G be a counterexample withas few edges as possible. If some end vertex of a path, say s1, has degreeone (with e = s1v the only edge), then we can embed G/e in U¨n for somen. Let G′ be obtained from U¨n by adding four vertices s′1, t ′1, s′2, t ′2, andedges s′1v1, s′1s′2, s′1 t ′2, s′2u1, s′2 t ′1, vn t ′1, un t ′2, t ′1 t ′2. Then G′ is isomorphic toU¨n+2, and G′ certainly has G as linkage minor.Hence we may assume that each end vertex of P1 and P2 has degreeat least two. Suppose no rung edge runs between two of these end ver-tices. Then it is not hard to see that G has an XX minor, a contradiction.Therefore some two end vertices must be connected. By reversing pathsas necessary, we may assume there is an edge e = s1s2.Since e crosses no edge, we can change our partition so that e ∈ B.By our assumption G\e can be embedded in U¨n for some n. Again, letG′ be obtained from U¨n by adding four vertices s′1, t ′1, s′2, t ′2, and edgess′1v1, s′1s′2, s′1 t ′2, s′2u1, s′2 t ′1, vn t ′1, un t ′2, t ′1 t ′2. Then G′ is isomorphic to U¨n+2,and G′ certainly has G as linkage minor, a contradiction.Now we have all ingredients of our main result.Proof of Theorem 1.2. From Lemma 2.2 we learn that (i)⇔(ii). FromLemma 2.3 we learn that (iii)⇒(ii), and from Lemmas 2.5 and 2.6 weconclude that (ii)⇒(iii).References[1] Carolyn Chun, Dillon Mayhew, Geoff Whittle, and Stefan H. M. vanZwam. The structure of binary Fano-fragile matroids. In preparation,2010.[2] Ken-ichi Kawarabayashi and Paul Wollan. A shorter proof of thegraph minor algorithm: the unique linkage theorem. In STOC, pages687–694, 2010.[3] Dillon Mayhew, Bogdan Oporowski, James Oxley, and Geoff Whit-tle. The excluded minors for the matroids that are binary or ternary.Submitted, 2009.[4] Neil Robertson and P. D. Seymour. Graph minors. XXI. graphs withunique linkages. J. Combin. Theory Ser. B, 99(3):583–616, 2009.6[5] K. Truemper. A decomposition theory of matroids. VI. Almost regu-lar matroids. J. Combin. Theory Ser. B, 55(2):235–301, 1992. ISSN0095-8956.7

The structure of graphs with a vital linkage of order 2

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