The proof of Theorem 4.0.2 is a simple upgrade of the k = 3 case (Oxley and Semple, 2013, Theorem 2.2).
Proof of Theorem 4.0.2. To prove the theorem, we show that k-Tree is a polynomial-time algorithm for finding a k-tree for M. Let T be the tree returned by a call tok-Tree(M). Then every vertex ofT is marked. More- over, by Proposition 6.2.5, T is a partial k-tree for M. Now T is a k-tree forM unless there is a non-sequentialk-separation ofM with the property that no equivalent k-separation is displayed by T. Suppose there is such a
k-separation (R, G). Since T is conforming, we may assume, by taking an equivalent k-separation if necessary, that G is contained in a bag B of T. IfT consists of the single bag vertex B, then line 3 ofk-Tree would have found a non-sequentialk-separation (Y, Z) ofM; a contradiction. But if T
consists of at least two vertices, then line 9 ofk-Tree would have found a non-sequentialk-separation (Y, Z) of M with the property that Z ⊆π(B), contradicting the fact thatB is marked. Hence T is ak-tree forM.
We next show that k-Tree runs in polynomial time in the size n of
E(M). By Lemma 5.1.1, the collectionFof maximal sequentialk-separating sets ofM can be constructed in polynomial time inn, and, by Theorem 5.1.2, for fixed disjoint subsets Y0 and Z0 of E(M), we can find a k-separation (Y, Z) with Y0 ⊆ Y and Z0 ⊆ Z in polynomial time in n, or determine
that none exists. Thus, by Lemma 5.1.3, we can find a non-sequential k- separation by iterating over all k-element subsets of E(M) not contained in a member of F. As there are O(nk) such subsets, wherek is fixed, this can be done in polynomial time in n. Extending this, whenever k-Tree, or one of the two subroutines, is called upon to find a k-separation where each part contains particular subsets, it either finds such ak-separation or correctly determines that there is no such k-separation in time polynomial inn. Therefore, as everyk-path ofM has lengthO(n), it follows that each call toForwardSweep takes time polynomial inn.
Now consider a call fromk-Tree to the subroutineBackwardSweep. When m ≥ 3, this subroutine considers each of the following subsets of
E(M) in turn: the subsets Zm and Zm−1, a subset Zi where i ∈ {m−2, m −3, . . . ,2}, and finally the subset X0 ∪ Z1. For each of the subsets
Z2, Z3, . . . , Zm−2, it is clear that their consideration takes polynomial time
inn. Note that finding the full closure of a subset X ofE(M), as in line 58 of BackwardSweep, takes time O(nk−1). For the subsets Zm and X0 ∪
Z1,BackwardSweepmay, up to five times, attempt to findk-separations
where each part contains particular subsets. As mentioned above, each call takes time polynomial in n, so the time taken for BackwardSweep to consider each of Zm and X0∪Z1 is also polynomial in n. Since m ≤n, it
follows that, whenm ≥ 3, BackwardSweep takes time polynomial in n. Similarly, the subroutine takes time polynomial in n when m = 2, so each call toBackwardSweeptakes time polynomial in n.
At the completion of each call to BackwardSweep, the algorithm k- Tree extends the current π-labelled tree to a new π-labelled tree in poly- nomial time inn. This extension is non-trivial in that at least one new edge is created. Since the terminal bags of each such constructed π-labelled tree contain at leastk−1 elements ofE(M) and there is no empty bag vertex of degree two, the number of edges of each constructedπ-labelled tree is linear inn, and so there areO(n) calls toForwardSweepandBackwardSweep from k-Tree. As marked bags are never reconsidered, we deduce that k- Treeterminates in time polynomial in n. This completes the proof of the theorem.
Bibliography
Aikin, J. M. (2009), The structure of 4-separations in 4-connected matroids, Ph.D. dissertation, Louisiana State University.
Aikin, J. and Oxley, J. (2008), ‘The structure of crossing separations in matroids’, Advances in Applied Mathematics41(1), 10–26.
Aikin, J. and Oxley, J. (2012), ‘The structure of the 4-separations in 4- connected matroids’, Advances in Applied Mathematics 48(1), 1–24. Beavers, B. D. (2006), Circuits and structure in matroids and graphs, Ph.D.
dissertation, Louisiana State University.
Bixby, R. E. (1982), ‘A simple theorem on 3-connectivity’, Linear Algebra and its Applications45, 123–126.
Brettell, N. and Semple, C. (2014a), ‘A Splitter Theorem relative to a fixed basis’,Annals of Combinatorics 18(1), 1–20.
Brettell, N. and Semple, C. (2014b), ‘An algorithm for constructing ak-tree for a k-connected matroid’, Annals of Combinatorics, to appear.
Brylawski, T. (1975), ‘Modular constructions for combinatorial geometries’,
Transactions of the American Mathematical Society 203, 1–44.
Chen, R. and Xiang, K. N. (2012), ‘Decomposition of 3-connected representable matroids’, Journal of Combinatorial Theory, Series B
102(3), 647–670.
Clark, B. and Whittle, G. (2013), ‘Tangles, trees, and flowers’, Journal of Combinatorial Theory, Series B103(3), 385–407.
Cook, W. J., Cunningham, W. H., Pulleyblank, W. R. and Schrijver, A. (1998), Matroid Intersection, in ‘Combinatorial Optimization’, Wiley, New York, pp. 287–294.
Coullard, C. and Oxley, J. (1992), ‘Extensions of Tutte’s wheels-and-whirls theorem’, Journal of Combinatorial Theory, Series B 56(1), 130–140. Cunningham, W. H. (1973), A combinatorial decomposition theory, Ph.D.
dissertation, University of Waterloo.
Cunningham, W. H. and Edmonds, J. (1980), ‘A combinatorial decomposi- tion theory’,Canadian Journal of Mathematics32(3), 734–765.
Edmonds, J. (1970), Submodular functions, matroids, and certain polyhe- dra, in ‘Combinatorial Structures and Their Applications’, Gordon and Breach, New York, pp. 69–87.
Geelen, J. F., Gerards, A. M. H. and Kapoor, A. (2000), ‘The excluded mi- nors forGF(4)-representable matroids’,Journal of Combinatorial Theory, Series B 79(2), 247–299.
Geelen, J. and Whittle, G. (2013), ‘Inequivalent representations of matroids over prime fields’, Advances in Applied Mathematics51(1), 1–175. Hall, R., Oxley, J. and Semple, C. (2005), ‘The structure of equivalent 3-
separations in a 3-connected matroid’, Advances in Applied Mathematics
35(2), 123–181.
Kahn, J. (1988), ‘On the uniqueness of matroid representations overGF(4)’,
Bulletin of the London Mathematical Society20, 5–10.
Oxley, J. (1996), ‘Structure theory and connectivity for matroids’,Contem- porary Mathematics197, 129–170.
Oxley, J. (2011),Matroid Theory, Vol. 21 ofOxford Graduate Texts in Math- ematics, second edition, Oxford University Press, New York.
Oxley, J. and Semple, C. (2013), ‘Constructing a 3-tree for a 3-connected matroid’,Advances in Applied Mathematics 50(1), 176–200.
Oxley, J., Semple, C. and Whittle, G. (2004), ‘The structure of the 3- separations of 3-connected matroids’, Journal of Combinatorial Theory, Series B 92(2), 257–293.
Oxley, J., Semple, C. and Whittle, G. (2008a), ‘Maintaining 3-connectivity relative to a fixed basis’, Advances in Applied Mathematics 41(1), 1–9. Oxley, J., Semple, C. and Whittle, G. (2008b), ‘Wild triangles in 3-connected
matroids’, Journal of Combinatorial Theory, Series B98(2), 291–323. Oxley, J., Semple, C. and Whittle, G. (2012), ‘An upgraded Wheels-and-
Whirls Theorem for 3-connected matroids’, Journal of Combinatorial Theory, Series B 102(3), 610–637.
Oxley, J., Vertigan, D. and Whittle, G. (1996), ‘On inequivalent represen- tations of matroids over finite fields’, Journal of Combinatorial Theory, Series B 67(2), 325–343.
Oxley, J. and Wu, H. (2000), ‘On the structure of 3-connected matroids and graphs’,European Journal of Combinatorics21(5), 667–688.
Seymour, P. D. (1980), ‘Decomposition of regular matroids’, Journal of Combinatorial Theory, Series B28(3), 305–359.
Seymour, P. D. (1995), Matroid Minors, in ‘Handbook of combinatorics’, R. Graham, M. Gr¨otschel and L. Lov´asz, editors, MIT press, Cambridge, pp. 527–550.
Tan, J. J.-M. (1981), Matroid 3-connectivity, Ph.D. dissertation, Carleton University.
Tutte, W. T. (1966), ‘Connectivity in matroids’,Canadian Journal of Math- ematics 18, 1301–1324.
Whittle, G. (1999), ‘Stabilizers of classes of representable matroids’,Journal of Combinatorial Theory, Series B 77(1), 39–72.
Whittle, G. and Williams, A. (2013), ‘On preserving matroid 3-connectivity relative to a fixed basis’, European Journal of Combinatorics34(6), 957– 967.