Tutte, W.T.: Graph Theory. In Encyclopedia of Mathematics and its Appl. Addison-Wesley, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as subgraph. If a and b are vertices in G, the length of any shortest path from a to b is called the distance between a and b in G, denoted by dc, a, b) or by d(a, b) if there is no possibility of confusion) H is called a minor of G if there is a subgraph of G which is contractible to H (see [40, 42, 75]) For natural numbers n 2 and m 0 define the n x n grid Q and the graph Q, as follows: V(Q) i,j) O i n and 0 j n , E(Q) i, j) i ,j ) li i l IJ J l 1 and O i, i ,j,j n , V(Q. i, j) 0 i (m 1) n 1) and 0 j (m 1) n 1) E(On,m) i, j) ....

....j) i , 0 i, i , j, j (m 1) n 1) li i l IJ i l 1 and ( i = 0 mod(m 1) and i=i ) or (j = 0 mod(m 1) and j =j ) Fig. 3 and Fig. 4 show Q7 and Q4,2 respectively. It is easy to verify the following fact by an induction on the size of the regarded graphs. Fact 1 (see [40, 75]) If H is planar, then there is a natural number n such that H is isomorphic to a minor of Q. There is a famous structure theory for graphs avoiding a given fixed graph as a minor, which was developed by Robertson and Seymour in their fundamental 7 Q4, 2 Fig. 3. Fig. 4. series of papers ....

W.T. Tutte, Graph Theory (Addison-Wesley, Reading, MA, 1984).

....The partition that lists the degrees of the vertices of the map is called the vertex partition. The partition that lists the degrees of the faces of the map is called the face partition, where the degree of a face is the number of edges that bound it. For further details the reader is referred to [25] and to the brief account given in [5] that is the starting point for this paper. Rooted maps occur in a number of contexts. These include the analysis of surfaces [20] the determination of the partition function [1] the determination of the reduced Euler characteristic [8] the generalisation ....

W.T.Tutte, "Graph theory", Encyclopedia of Math. and its Applications 21, Addison-Wesley, London, 1984. 19

....descriptions of neighborhood relations, boundaries, and so on. Meanwhile, a number of different methods for the representation of finite topologies has been proposed, including cellular complexes [9] block complexes [10] the star topology [1] the Khalimsky grid [6, 3] combinatorial maps [15, 5] and border maps [2] All methods approach the problem somewhat differently, but they have a lot in common as well. Unfortunately, the commonalties are not immediately apparent from the literature because most authors present their method in isolation or emphasize the differences to other methods. ....

....of smaller dimension, and the relation must be transitive. This means that all 1 blocks must be sequences of adjacent 0 and 1 cells, and all junctions between 1 blocks must be O blocks. Another approach to finite topological spaces originates from the field of graph theory the combinatorial map [15, 5]: Definition 4: A combinatorial map is a triple (D, or, c 0 where D is a set of darts (also known as half edges) cr is a permutation of the darts, and c is an involu tion of the darts. In this context, a permutation is a mapping that associates to each dart a unique predecessor and a unique ....

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W.T. Tutte: "Graph Theory", Cambridge University Press, 1984

....descriptions of neighborhood relations, boundaries, and so on. Meanwhile, a number of different methods for the representation of finite topologies has been proposed, including cellular complexes [9, 7] block complexes [10] the star topology [1] the Khalimsky grid [6, 3] combinatorial maps [16, 5] and border maps [2] All methods approach the problem somewhat differently, but they have a lot in common as well. Unfortunately, the commonalties are not immediately apparent from the literature because most authors present their method in isolation or emphasize the differences to other methods. ....

....This poses some restrictions on valid partitions. In particular, all 1 blocks must be sequences of adjacent 0 and 1 cells, and all junctions between 1 blocks must be O blocks. Another approach to finite topological spaces originates from the field of graph theory the combinatorial map [16, 5]: Definition 4: A combinatorial map is a triple (D, or, c 0 where D is a set of darts (also known as half edges) cr is a permutation of the darts, and c is an involution of the darts. In this context, a permutation is a mapping that associates to each dart a umque predecessor and a umque ....

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W.T. Tutte: "Graph Theory", Cambridge University Press, 1984

....called faces. Maps are considered up to homeomorphisms of the surface taking vertices to vertices, edges to edges and faces to faces. Like graphs, maps with a finite number of edges can be given various combinatorial representations. We shall use the cross representation, introduced by Tutte [9]: each edge is viewed as a fat ribbon with four corners or crosses (see Figure 1) and three fix point free involutions are defined on the set A c of crosses. The first two, fl and ffi, relate two by two the four adjacent crosses of each edge, fl along faces and ffi along vertices. The last one, ....

Tutte (W. T.). -- Graph theory. -- Addison-Wesley Publishing Co., Reading, Mass., 1984, Encyclopedia of Mathematics and its Applications, vol. 21, xxi+333p. With a foreword by C. St. J. A. Nash-Williams.

....These results extend to nonorientable surfaces: the main di erence comes from the observation that the global orientation can not be used anymore to de ne a permutation on the set of darts around a vertex. However, a similar structure can be de ned only from the local orientation (see [11]) which suces to our purpose. Acknowledgements We thank G. Schae er for helpful remarks. ....

W.T. Tutte. Graph theory. Addison Wesley, 1984.

....subset of E(H) If Q is a full contraction of H with respect to S, then n(Q) 60jE(H) Gamma Sj. Proof. Let C denote the set E(H) Gamma S and let Q 0 denote the graph Q Gamma C. To bound n(Q) we consider the block graph of Q 0 . Let H 0 be an arbitrary graph. The block graph of H 0 ([23]) denoted by blk(H 0 ) is a bipartite graph whose vertices are the cutpoints and blocks of H 0 . A block is connected in blk(H 0 ) to exactly those cutpoints that it contains in H 0 . It is known ( 23] that the block graph of H 0 is a tree for any connected graph H 0 . Thus the ....

....the block graph of Q 0 . Let H 0 be an arbitrary graph. The block graph of H 0 ( 23] denoted by blk(H 0 ) is a bipartite graph whose vertices are the cutpoints and blocks of H 0 . A block is connected in blk(H 0 ) to exactly those cutpoints that it contains in H 0 . It is known ([23]) that the block graph of H 0 is a tree for any connected graph H 0 . Thus the graph blk(Q 0 ) is a forest. Let n 0 and n 1 denote the number of degree 0 and degree 1 nodes in blk(Q 0 ) If n(blk(Q 0 ) 1, then the graph (V (H) S) has a unique block and this block contains at least 3 ....

W. Tutte, Graph theory, Addison-Wesley, 1984.

....minor of a graph G , if H can be obtained from a (not necessarily induced) subgraph of G by contracting edges. Contracting an edge (a 1 ; a 2 ) 2 E(G ) means removing the vertices a 1 ; a 2 , adding a new vertex b, and replacing all edges (u; a i ) i = 1; 2, by (u; b) for more details, see e.g. [Tut84]. A class C is de ned by forbidden minors H 1 ; H n , C = FORB(H 1 ; H n ) if for every graph G , G 2 C if and only if no H i is a minor of G , for 1 i n. 17 In [Nur96] the following notion was used to classify certain classes of graphs de ned by forbidden minors. 4.6 ....

....has non trivial minors. Thus we conclude that for all e, G 0 e 2 C and G e 62 C. The claim follows now from Corollary 3.7 and Theorem 4.2. 2 Especially Theorem 4.8 holds for the class PLAN of planar graphs and the class ACYC of acyclic graphs. This is because PLAN = FORB(K 5 ; K 3;3 ) see e.g. [Tut84]) and ACYC = FORB(K 3 ) 4.9 Corollary. For every positive integer k, the k th extended vectorizations of Q PLAN and QACYC are not de nable in FO(Q k ) Thus the extended vectorizations of FO(Q PLAN ) and FO(QACYC ) are not nitely generated. 5 Conclusion We conclude the paper by considering a ....

W. Tutte. Graph Theory. Addison-Wesley, 1984.

....and z be three distinct vertices of attachment of a bridge B of J in G. Then there is a vertex v belonging to the nucleus of B for which there are three internally vertex disjoint paths in B: Y 1 [x; v] Y 2 [y : v] and Y 3 [z; v] Section 3 Overview of the Algorithm 9 Remark 2. 16 Following Tutte[22], we define a Y graph as the union Y of three paths Y 1 , Y 2 and Y 3 which have one end v in common but are otherwise mutually disjoint. We call v the center and the paths Y i s, the arms of Y . Proposition 2.5 Let x, y and z be three distinct vertices of attachment of bridge B of J in G and ....

W.T. Tutte. Graph Theory. Addison-Wesley Publishing Co., Menlo Park, California, 1984.

....4 5 6 Figure 4. An edge sum tree T and its composition G(T) from the above proposition and Lemma 4.1 that composing a block tree produces a unique (unlabeled) 2 connected graph. It is natural to ask whether every 2connected graph has a decomposition into some kind of block tree. Indeed, Tutte [7] proved the following: 4.2. Theorem. If G is a 2 connected graph containing at least three edges, then it can be decomposed into a 3 block tree. Moreover, this decomposition is unique (up to equivalence of 3 block trees) Later, we shall use the existence of such a decomposition guaranteed by ....

W. T. Tutte, Graph theory, Addison-Wesley, Menlo Park, California, 1984. (Dittmann, Oporowski) Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918, USA E-mail address, Dittmann, Oporowski: [dittmann, bogdan]@math.lsu.edu

....of the types of M 1 and M 2 . Whenever we speak of a submap, we will assume that the face formed by the added disc is distinguished, and, therefore, so too are the edges of the cycle C. There are several definitions of k connectivity in the literature. We will use a variant of Tutte s definition [33]. The definition used here is from Graver and Watkins [24] and on graphs with at least three edges is equivalent to Tutte s definition. Definition. A graph G is k connected if its girth (or shortest cycle size) is at least k and at least k vertices must be removed to separate the graph. Thus, ....

....its dual. For our purposes, this representation is inadequate on two counts. First, it does not work for maps on a non orientable surface. Second, on an orientable surface it does not give a unique representation: hA d ; fl; aei and hA d ; fl; ae Gamma1 i represent the same map. Tutte [31, 33] extended Edmond s representation to overcome some of these difficulties. He used crosses rather than darts. A cross is an edge with a direction and designated side. A cross is determined by a dart in the map and a crossing dart the dual map. The dart in the dual map points to the designated side ....

William T. Tutte. Graph Theory. Addison-Wesley, Reading, MA, 1984. 24 BENDER, COMPTON AND RICHMOND

....Vertices The combinatorial structure of a decomposition star is conveniently described as a planar map. A planar graph is a graph that can be embedded into the plane or sphere. A planar map is a planar graph with additional combinatorial structure that encodes a particular embedding of the graph [T]. All our planar maps will be unoriented: we do not distinguish between a planar map and its reflection. Associated with a planar map are faces, combinatorial) angles between adjacent edges, and so forth. Associated with each planar map L is a planar graph G(L) obtained by forgetting the ....

. W.T. Tutte, Graph theory, Addison-Wesley, 1984. SPHERE PACKINGS III 17

....in every degree. Another, and probably more important application of the algebra W is that it yields a series of pre invariants for chord diagrams (see [CDL1] through the notion of weighted chromatic invariants. This construction is an analog of the classical theory of Tutte invariants (see [T]) adjusted for graphs 5 without multiple edges and loops, but having weighted vertices. Example. Let D be a chord diagram. Consider the mapping : D 7 ( Gamma(D) where Gamma(D) is the intersection graph of the chord diagram D, and ( Gamma(D) 2 Z[t] its chromatic polynomial. The mapping ....

....theory of Tutte invariants (see [T] adjusted for graphs 5 without multiple edges and loops, but having weighted vertices. Example. Let D be a chord diagram. Consider the mapping : D 7 ( Gamma(D) where Gamma(D) is the intersection graph of the chord diagram D, and ( Gamma(D) 2 Z[t] its chromatic polynomial. The mapping determines a preinvariant : M Z[t] with values in the ring of one variable polynomials. 1.3 Weighted graphs and chromatic relations Definition. A weighted graph is a graph G without loops and multiple edges given together with a mapping w : V (G) N ....

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W. T. Tutte. Graph Theory, Addison-Wesley, 1984.

....properties [Cou10] a depth first spanning tree ( Cou8, Cou14] just to take a few examples. Certain graphs have a unique such structure: for instance, every graph has a unique modular decomposition, every 3 connected graph has a unique decomposition in 3 connected components, a result by Tutte [Tu] used in [Cou11] every planar 3 connected graph has a unique planar representation, by a theorem of Whitney (used in [Cou12] It seems that structures are easier to define by MS formulas when they are unique. The constructions are otherwise more difficult [Lapoire] or impossible (for instance ....

W Tutte, Graph Theory, Addison Wesley 1984.

....above. As a second type of examples, consider classes of graphs defined by forbidden minors. A graph H is a minor of a graph G , if H can be obtained from G by contracting edges (an edge between two vertices is collapsed to length zero) or deleting isolated vertices (for more details, see e.g. [Tut84]) A class C is defined by forbidden minors H 1 ; H n , C = FORB(H 1 ; H n ) if for every graph G , G 2 C if and only if no H i is a minor of G , for 1 i n. Obviously for any such class C, both C and C are closed under stretching. In [Nur96] the following notion was used to ....

....be trivial. The proof in [Nur96] shows that the countergraphs can be selected to be of color class size two. Hence we can apply Corollary 3.7. Especially Theorem 4.8 holds for the class PLAN of planar graphs and the class ACYC of acyclic graphs. This is because PLAN = FORB(K 5 ; K 3;3 ) see e.g. [Tut84]) and ACYC = FORB(K 3 ) 4.9 Corollary. For every positive integer k, the k th extended vectorizations of QPLAN and QACYC are not definable in FO(Q k ) Thus the extended vectorizations of FO(QPLAN ) and FO(QACYC ) are not finitely generated. 4.3 Applications for FV(Q k ) So far we have not ....

W. Tutte. Graph Theory. Addison-Wesley, 1984.

....of B. Proposition 2.4 Let x, y and z be three distinct vertices of attachment of a bridge B of J in G. Then there is a vertex v belonging to the nucleus of B for which there are three internally vertex disjoint paths in B: Y 1 [x; v] Y 2 [y : v] and Y 3 [z; v] Remark 2. 16 Following Tutte[21], we define a Y graph as the union Y of three paths Y 1 , Y 2 and Y 3 which have one end v in common but are otherwise mutually disjoint. We call v the center and the paths Y i s, the arms of Y . Proposition 2.5 Let x, y and z be three distinct vertices of attachment of bridge B of J in G and ....

W.T. Tutte. Graph Theory. Addison-Wesley Publishing Co., Menlo Park, California, 1984.

....structure of a Delaunay star is conveniently described as a planar map. A planar graph is a graph that can be embedded into the plane or sphere. A 14 5. TYPES OF VERTICES planar map is a planar graph with additional combinatorial structure that encodes a particular embedding of the graph [T]. All our planar maps will be unoriented: we do not distinguish between a planar map and its reflection. Associated with a planar map are faces, combinatorial) angles between adjacent edges, and so forth. Associated with each planar map L is a planar graph G(L) obtained by forgetting the ....

. W.T. Tutte, Graph theory, Addison-Wesley, 1984.

....on any B i except possibly c 1 and c k 1 are incident with a redundant edge of H. A maximal block chain in H is a block chain in H not properly contained in any other block chain of H. It is helpful to interpret block chains in an auxiliary graph which we shall now define. The block graph of H ([14]) denoted by blk(H) is a bipartite graph whose vertices are the cutpoints and blocks of H. A block is connected in blk(H) to exactly those cutpoints that it contains in H. It is known that the block graph of H is a tree for any connected graph H. Now consider a biconnected graph H. Define a ....

W. Tutte, Graph theory, Addison-Wesley, 1984.

....defined by means of MS formulas. The unique decomposition of a cograph is definable in this way by MS formulas, provided the graph is given with an auxiliary linear ordering [6] In the present paper, we investigate in this perspective the unique decomposition of connected graphs defined by Tutte [17] and we prove that it is definable by MS formulas. This decomposition has two levels : every connected graph is a tree of 2 connected components called blocks ; every 2 connected graph is a tree of so called 3 blocks, which are some kinds of subgraphs of the given graph. Our proof uses 2 dags ....

....smaller with respect to than the smallest edge of g(z) This defines a linear order because the sets E(g(y) and E(g(z) are disjoint and this order is MS definable in the structure (G,T,g) 2 augmented with . 4. The Tutte decomposition of a 2 connected graph We recall the result of Tutte [17] stating that a 2 connected undirected graph G has a unique decomposition in so called 3 blocks . This decomposition can be obtained from the canonical decomposition of any 2 dag H such that und(H) G, 23 as we shall see. From Theorem 3.12, it will follow that the Tutte decomposition of G is ....

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TUTTE W., Graph Theory, Addison-Wesley, 1984.

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Tutte, W.T.: Graph Theory. In Encyclopedia of Mathematics and its Appl. Addison-Wesley, 1984.

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W.T. Tutte. Graph theory. Addison Wesley, 1984.

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W.T. Tutte. Graph theory. Addison Wesley, 1984.

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W. Tutte. Graph Theory. Addison-Wesley, 1984. 22

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W.T. Tutte,1984. Graph Theory. Addison-Wesley Publishing Company.

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W. T. Tutte. Graph Theory. Addison-Wesley, Menlo Park, CA., 1984.

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