607:
31:
578:
for graphs, but so far this has been proven only for the matroids of bounded branchwidth. Additionally, if a minor-closed family of matroids representable over a finite field does not include the graphic matroids of all planar graphs, then there is a constant bound on the branchwidth of the matroids
500:
that generalizes branch-decompositions of graphs. A branch-decomposition of a matroid is a hierarchical clustering of the matroid elements, represented as an unrooted binary tree with the elements of the matroid at its leaves. An e-separation may be defined in the same way as for graphs, and results
699:
and the non-graphic matroid U(2,4). The matroids of branchwidth three are not well-quasi-ordered without the additional assumption of representability over a finite field, but nevertheless the matroids with any finite bound on their branchwidth have finitely many minimal forbidden minors, all of
377:
Carving width is a concept defined similarly to branch width, except with edges replaced by vertices and vice versa. A carving decomposition is an unrooted binary tree with each leaf representing a vertex in the original graph, and the width of a cut is the number (or total weight in a weighted
545:
have different branchwidths, 2 and 1 respectively, but they both induce the same graphic matroid with branchwidth 1. However, for graphs that are not trees, the branchwidth of the graph is equal to the branchwidth of its associated graphic matroid. The branchwidth of a matroid is equal to the
487:
argue that branchwidth works better than treewidth in the development of fixed-parameter-tractable algorithms on planar graphs, for multiple reasons: branchwidth may be more tightly bounded by a function of the parameter of interest than the bounds on treewidth, it can be computed exactly in
686:
Forbidden minors have also been studied for matroid branchwidth, despite the lack of a full analogue to the
RobertsonâSeymour theorem in this case. A matroid has branchwidth one if and only if every element is either a loop or a coloop, so the unique minimal forbidden minor is the
565:
can also be generalized to matroids, and plays a bigger role than branchwidth in the theory of graph minors, branchwidth has more convenient properties in the matroid setting. Robertson and
Seymour conjectured that the matroids representable over any particular
646:; the minimal forbidden minors for branchwidth 1 are the triangle graph (or the two-edge cycle, if multigraphs rather than simple graphs are considered) and the three-edge path graph. The graphs of branchwidth 2 are the graphs in which each
361:
440:, the branchwidth can be computed exactly in polynomial time. This in contrast to treewidth for which the complexity on planar graphs is a well known open problem. The original algorithm for planar branchwidth, by
691:
U(2,3), the graphic matroid of the triangle graph. A matroid has branchwidth two if and only if it is the graphic matroid of a graph of branchwidth two, so its minimal forbidden minors are the graphic matroid of
131:
is a connected undirected graph with no cycles in which each non-leaf node has exactly three neighbors. A branch-decomposition may be represented by an unrooted binary tree
533:, and the width of the decomposition and the branchwidth of the matroid are defined analogously. The branchwidth of a graph and the branchwidth of the corresponding
107:. And as with treewidth, many graph optimization problems may be solved efficiently for graphs of small branchwidth. However, unlike treewidth, the branchwidth of
638:
and a triangle graph (or the two-edge cycle, if multigraphs rather than simple graphs are considered). The graphs of branchwidth 1 are the graphs in which each
471:
algorithms for many NP-hard optimization problems, using an amount of time that is exponential in the width of the input graph or matroid. For instance,
1389:
972:. The fourth forbidden minor for treewidth three, the pentagonal prism, has the cube graph as a minor, so it is not minimal for branchwidth three.
84:
into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The
302:
1398:
274:: the two quantities are always within a constant factor of each other. In particular, in the paper in which they introduced branch-width,
1232:
668:
as a minor; therefore, the four minimal forbidden minors are three of the four forbidden minors for treewidth three (the graph of the
1218:
381:
Branch width algorithms typically work by reducing to an equivalent carving width problem. In particular, the carving width of the
103:: for all graphs, both of these numbers are within a constant factor of each other, and both quantities may be characterized by
1324:
1030:
802:
Another long-standing open problem is whether there is a polynomial-time algorithm to compute the treewidth of planar graphs.
1098:
Fomin, Fedor V.; Thilikos, Dimitrios M. (2006), "Dominating sets in planar graphs: branch-width and exponential speed-up",
550:, and in particular this implies that the branchwidth of any planar graph that is not a tree is equal to that of its dual.
1465:
Mazoit, Frédéric; Thomassé, Stéphan (2007), "Branchwidth of graphic matroids", in Hilton, Anthony; Talbot, John (eds.),
639:
795:
483:
heuristic to find a good branch-decomposition of this graph, and applying dynamic programming to the decomposition.
488:
polynomial time rather than merely approximated, and the algorithm for computing it has no large hidden constants.
104:
1489:
425:: there is an algorithm for computing optimal branch-decompositions whose running time, on graphs of branchwidth
1520:
1150:
275:
1604:
1584:
619:
575:
250:. The width of the branch-decomposition is the maximum width of any of its e-separations. The branchwidth of
664:
on four vertices. A graph has branchwidth three if and only if it has treewidth three and does not have the
1550:
1524:
1484:
1067:
445:
441:
279:
479:
into a single global solution, by forming a sparse graph from the union of the partial solutions, using a
1589:
1474:, London Mathematical Society Lecture Note Series, vol. 346, Cambridge University Press, p. 275
476:
17:
1466:
1210:
1183:; Gerards, Bert; Whittle, Geoff (2002), "Branch-width and well-quasi-ordering in matroids and graphs",
1345:
Hicks, Illya V.; McMurray, Nolan B. Jr. (2007), "The branchwidth of graphs and their cycle matroids",
774:), at the expense of an increase in the dependence on the number of vertices from linear to quadratic.
475:
apply branchwidth-based dynamic programming to a problem of merging multiple partial solutions to the
1594:
651:
1421:
422:
1599:
631:
58:
38:, showing an e-separation. The separation, the decomposition, and the graph all have width three.
456:
vertices, and their algorithm for constructing a branch decomposition of this width took time O(
1259:
1271:
Gu, Qian-Ping; Tamaki, Hisao (July 2008), "Optimal branch-decomposition of planar graphs in O(
785:
1368:
Proc. 28th
International Symposium on Mathematical Foundations of Computer Science (MFCS '03)
1071:
647:
1512:
418:
413:
are both considered as inputs to the problem. However, the graphs with branchwidth at most
128:
66:
8:
643:
571:
542:
480:
468:
1370:, Lecture Notes in Computer Science, vol. 2747, Springer-Verlag, pp. 470â479,
1009:, Lecture Notes in Computer Science, vol. 1256, Springer-Verlag, pp. 627â637,
1005:; Thilikos, Dimitrios M. (1997), "Constructive linear time algorithms for branchwidth",
1040:
1002:
267:
100:
1171:
1007:
Proc. 24th
International Colloquium on Automata, Languages and Programming (ICALP '97)
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1251:
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51:
1153:; Whittle, Geoff (2003), "On the excluded minors for the matroids of branch-width
1508:
688:
665:
627:
611:
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591:
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112:
1120:
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which have a number of elements that is at most exponential in the branchwidth.
1503:
1358:
1300:; Semple, Charles; Whittle, Geoff (2002), "On matroids of branch-width three",
1255:
655:
595:
115:. Branch-decompositions and branchwidth may also be generalized from graphs to
1454:
1417:
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1111:
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in the family, generalizing similar results for minor-closed graph families.
558:
372:
236:; that is, it is the number of vertices that are shared by the two subgraphs
185:
into subtrees induces a partition of the edges associated with the leaves of
108:
1288:
1314:
1197:
1057:
680:
567:
547:
514:
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of a planar graph is exactly twice the branch width of the original graph.
382:
356:{\displaystyle b-1\leq k\leq \left\lfloor {\frac {3}{2}}b\right\rfloor -1.}
43:
1366:
HlinÄnĂœ, Petr (2003), "On matroid properties definable in the MSO logic",
1043:; Thilikos, Dimitrios M. (1999), "Graphs with branchwidth at most three",
553:
Branchwidth is an important component of attempts to extend the theory of
1480:
1297:
554:
394:
1440:
HlinÄnĂœ, Petr; Whittle, Geoff (2009), "Addendum to matroid tree-width",
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35:
30:
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It is also possible to define a notion of branch-decomposition for
497:
116:
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of the matroid, then the width of an e-separation is defined as
378:
graph) of edges that are incident to a vertex in both subtrees.
1527:(1991), "Graph minors. X. Obstructions to tree-decomposition",
1121:"Computing branchwidth via efficient triangulations and blocks"
1145:
985:
1553:; Thomas, Robin (1994), "Call routing and the ratcatcher",
467:
As with treewidth, branchwidth can be used as the basis of
421:, from which it follows that computing the branchwidth is
218:
The width of an e-separation is the number of vertices of
1119:
Fomin, Fedor V.; Mazoit, Frédéric; Todinca, Ioan (2009),
266:
Branch-decompositions of graphs are closely related to
1295:
1211:"Towards a structure theory for matrices and matroids"
981:
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by an algorithm that has access to the matroid via an
305:
1227:
1205:
1179:
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is the minimum width of any branch-decomposition of
807:
784:Kao, Ming-Yang, ed. (2008), "Treewidth of graphs",
1118:
1039:
1001:
969:
844:. Section 12, "Tangles and Matroids", pp. 188â190.
763:describe an algorithm with improved dependence on
760:
756:
355:
254:is the minimum width of a branch-decomposition of
135:, together with a bijection between the leaves of
1576:
1519:
957:
841:
744:
715:
1329:, Ph.D. thesis, Rice University, archived from
1464:
1439:
1396:
1388:: CS1 maint: DOI inactive as of August 2024 (
1219:Proc. International Congress of Mathematicians
884:
868:
856:
634:; the minimal forbidden minors are a two-edge
1549:
1344:
1097:
872:
732:
491:
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1326:Branch Decompositions and their Applications
433:, is linear in the size of the input graph.
401:has a branch-decomposition of width at most
388:
1065:
953:
951:
949:
472:
1479:
940:
654:; the only minimal forbidden minor is the
1540:
1502:
1453:
1347:Journal of Combinatorial Theory, Series B
1313:
1302:Journal of Combinatorial Theory, Series B
1244:Journal of Combinatorial Theory, Series B
1196:
1185:Journal of Combinatorial Theory, Series B
1170:
1159:Journal of Combinatorial Theory, Series B
1136:
1056:
537:may differ: for instance, the three-edge
270:, and branch-width is closely related to
1270:
1231:; Gerards, Bert; Whittle, Geoff (2007),
1209:; Gerards, Bert; Whittle, Geoff (2006),
946:
813:
626:can be characterized by a finite set of
605:
586:, the matroids with branchwidth at most
261:
29:
1365:
1072:"Tour merging via branch-decomposition"
829:
14:
1577:
1397:HlinÄnĂœ, Petr; Whittle, Geoff (2006),
852:
850:
728:
726:
724:
630:. The graphs of branchwidth 0 are the
76:as its leaves. Removing any edge from
27:Hierarchical clustering of graph edges
1322:
825:
505:of matroid elements into two subsets
222:that are incident both to an edge of
929:Geelen, Gerards & Whittle (2007)
925:Geelen, Gerards & Whittle (2006)
913:Geelen, Gerards & Whittle (2006)
909:Geelen, Gerards & Whittle (2002)
897:Geelen, Gerards & Whittle (2006)
847:
783:
721:
601:
24:
1233:"Excluding a planar graph from GF(
761:Fomin, Mazoit & Todinca (2009)
99:Branchwidth is closely related to
25:
1616:
1442:European Journal of Combinatorics
1406:European Journal of Combinatorics
1222:, vol. III, pp. 827â842
139:and the edges of the given graph
1487:(2007), "Testing branch-width",
970:Bodlaender & Thilikos (1999)
757:Bodlaender & Thilikos (1997)
683:) together with the cube graph.
614:for graphs of branchwidth three.
366:
211:into two subgraphs is called an
167:partitions it into two subtrees
1529:Journal of Combinatorial Theory
1490:Journal of Combinatorial Theory
975:
963:
934:
918:
902:
890:
878:
862:
460:). This was later sped up to O(
1277:ACM Transactions on Algorithms
958:Robertson & Seymour (1991)
835:
819:
777:
750:
745:Robertson & Seymour (1991)
738:
709:
122:
13:
1:
1468:Surveys in Combinatorics 2007
1376:10.1007/978-3-540-45138-9\_41
1172:10.1016/S0095-8956(02)00046-1
994:
419:minor-closed family of graphs
397:to determine whether a graph
1542:10.1016/0095-8956(91)90061-N
1125:Discrete Applied Mathematics
1079:INFORMS Journal on Computing
885:HlinÄnĂœ & Whittle (2006)
869:Mazoit & Thomassé (2007)
857:Mazoit & Thomassé (2007)
842:Robertson & Seymour 1991
716:Robertson & Seymour 1991
622:, the graphs of branchwidth
111:may be computed exactly, in
7:
1091:10.1287/ijoc.15.3.233.16078
873:Hicks & McMurray (2007)
733:Seymour & Thomas (1994)
485:Fomin & Thilikos (2006)
477:travelling salesman problem
10:
1621:
1504:10.1016/j.jctb.2006.06.006
1438:Addendum and corrigendum:
1359:10.1016/j.jctb.2006.12.007
1256:10.1016/j.jctb.2007.02.005
787:Encyclopedia of Algorithms
501:in a partition of the set
492:Generalization to matroids
370:
34:Branch decomposition of a
1455:10.1016/j.ejc.2008.09.028
1418:10.1016/j.ejc.2006.06.005
1237:)-representable matroids"
1138:10.1016/j.dam.2008.08.009
1112:10.1137/S0097539702419649
1100:SIAM Journal on Computing
1015:10.1007/3-540-63165-8_217
790:, Springer, p. 969,
620:RobertsonâSeymour theorem
576:RobertsonâSeymour theorem
473:Cook & Seymour (2003)
423:fixed-parameter tractable
389:Algorithms and complexity
1323:Hicks, Illya V. (2000),
941:Oum & Seymour (2007)
703:
282:showed that for a graph
155:is any edge of the tree
80:partitions the edges of
1289:10.1145/1367064.1367070
582:For any fixed constant
429:for any fixed constant
59:hierarchical clustering
1315:10.1006/jctb.2002.2120
1198:10.1006/jctb.2001.2082
1058:10.1006/jagm.1999.1011
960:, Theorem 4.2, p. 165.
814:Gu & Tamaki (2008)
747:, Theorem 4.1, p. 164.
718:, Theorem 5.1, p. 168.
615:
357:
39:
1378:(inactive 2024-08-12)
1045:Journal of Algorithms
672:, the complete graph
652:seriesâparallel graph
648:biconnected component
609:
590:can be recognized in
574:, analogously to the
358:
262:Relation to treewidth
33:
1605:NP-complete problems
1585:Trees (graph theory)
1399:"Matroid tree-width"
986:Geelen et al. (2003)
303:
207:. This partition of
181:. This partition of
129:unrooted binary tree
67:unrooted binary tree
65:, represented by an
48:branch-decomposition
1041:Bodlaender, Hans L.
1003:Bodlaender, Hans L.
640:connected component
596:independence oracle
546:branchwidth of its
541:and the three-edge
513:. If Ï denotes the
481:spectral clustering
469:dynamic programming
268:tree decompositions
189:into two subgraphs
1590:Graph minor theory
1567:10.1007/BF01215352
982:Hall et al. (2002)
616:
572:well-quasi-ordered
353:
229:and to an edge of
72:with the edges of
40:
1283:(3): 30:1â30:13,
1149:; Gerards, Bert;
1131:(12): 2726â2736,
1032:978-3-540-63165-1
452:) on graphs with
337:
16:(Redirected from
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1595:Graph invariants
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1551:Seymour, Paul D.
1545:
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159:, then removing
105:forbidden minors
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52:undirected graph
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656:complete graph
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559:matroid minors
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448:, took time O(
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371:Main article:
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276:Neil Robertson
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1555:Combinatorica
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1485:Seymour, Paul
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1427:on 2012-03-06
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1333:on 2011-07-16
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1265:on 2010-09-24
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