Euler’s Algorithm
2.6 Planar Duality
We shall now introduce an important duality concept. This is the only place in this book where we need loops. So in this section loops, i.e. edges whose endpoints coincide, are allowed. In a planar embedding loops are of course represented by polygons instead of polygonal arcs.
Note that Euler’s formula (Theorem 2.32) also holds for graphs with loops: this follows from the observation that subdividing a loope (i.e. replacing e= {v, v}
by two parallel edges{v, w},{w, v}wherew is a new vertex) and adjusting the embedding (replacing the polygon Je by two polygonal arcs whose union is Je) increases the number of edges and vertices each by one but does not change the number of faces.
Definition 2.41. LetGbe a directed or undirected graph, possibly with loops, and let =(ψ, (Je)e∈E(G))be a planar embedding ofG. We define theplanar dual G∗whose vertices are the faces of and whose edge set is{e∗:e∈ E(G)}, where e∗ connects the faces that are adjacent to Je (if Je is adjacent to only one face, thene∗is a loop). In the directed case, say fore=(v, w), we oriente∗=(F1,F2) in such a way that F1 is the face “to the right” when traversing Je fromψ(v)to ψ(w).
G∗ is again planar. In fact, there obviously exists a planar embedding ψ∗, (Je∗)e∗∈E(G∗)
of G∗ such that ψ∗(F) ∈ F for all faces F of and, for eache∈E(G),|Je∗∩Je| =1 and
Je∗∩
⎛
⎝{ψ(v):v∈V(G)} ∪
f∈E(G)\{e}
Jf
⎞
⎠ = ∅.
Such an embedding is called astandard embeddingofG∗.
2.6 Planar Duality 41
(a) (b)
Fig. 2.8.
The planar dual of a graph really depends on the embedding: consider the two embeddings of the same graph shown in Figure 2.8. The resulting planar duals are not isomorphic, since the second one has a vertex of degree four (corresponding to the outer face) while the first one is 3-regular.
Proposition 2.42. Let G be an undirected connected planar graph with a fixed embedding. LetG∗be its planar dual with a standard embedding. Then(G∗)∗=G.
Proof: Let
ψ, (Je)e∈E(G)
be a fixed embedding ofGand
ψ∗, (Je∗)e∗∈E(G∗) a standard embedding of G∗. Let F be a face of G∗. The boundary of F contains Je∗ for at least one edge e∗, so F must containψ(v)for one endpoint v ofe. So every face of G∗ contains at least one vertex ofG.
By applying Euler’s formula (Theorem 2.32) toG∗ and to G, we get that the number of faces ofG∗ is|E(G∗)| − |V(G∗)| +2= |E(G)| −(|E(G)| − |V(G)| + 2)+2= |V(G)|. Hence each face ofG∗ contains exactly one vertex ofG. From this we conclude that the planar dual ofG∗ is isomorphic toG. 2 The requirement thatG is connected is essential here: note thatG∗ is always connected, even ifG is disconnected.
Theorem 2.43. Let G be a connected planar undirected graph with arbitrary embedding. The edge set of any circuit inGcorresponds to a minimal cut in G∗, and any minimal cut inG corresponds to the edge set of a circuit inG∗.
Proof: Let =(ψ, (Je)e∈E(G))be a fixed planar embedding ofG. LetC be a circuit inG. By Theorem 2.30,R2\
e∈E(C)Jesplits into exactly two connected regions. Let A and B be the set of faces of in the inner and outer region, respectively. We have V(G∗)=A∪. B andEG∗(A,B)= {e∗ :e∈ E(C)}. Since
Aand B form connected sets inG∗, this is indeed a minimal cut.
Conversely, letδG(A)be a minimal cut inG. Let ∗ =(ψ∗, (Je)e∈E(G∗))be a standard embedding ofG∗. Leta∈ A andb∈V(G)\A. Observe that there is no polygonal arc in
R:=R2\
⎛
⎝{ψ∗(v):v∈V(G∗)} ∪
e∈δG(A)
Je∗
⎞
⎠
which connects ψ(a)and ψ(b): the sequence of faces of G∗ passed by such a polygonal arc would define an edge progression from a tob in G not using any edge ofδG(A).
SoRconsists of at least two connected regions. Then, obviously, the boundary of each region must contain a circuit. Hence F := {e∗ :e∈ δG(A)}contains the edge set of a circuitC inG∗. We have{e∗ :e∈E(C)} ⊆ {e∗:e∈ F} =δG(A), and, by the first part, {e∗ : e ∈ E(C)} is a minimal cut in (G∗)∗ = G (cf.
Proposition 2.42). We conclude that{e∗ :e∈E(C)} =δG(A). 2 In particular, e∗ is a loop if and only if e is a bridge, and vice versa. For digraphs the above proof yields:
Corollary 2.44. Let G be a connected planar digraph with some fixed planar embedding. The edge set of any circuit inGcorresponds to a minimal directed cut
inG∗, and vice versa. 2
Another interesting consequence of Theorem 2.43 is:
Corollary 2.45. Let G be a connected undirected graph with arbitrary planar embedding. ThenG is bipartite if and only ifG∗ is Eulerian, andGis Eulerian if and only ifG∗is bipartite.
Proof: Observe that a connected graph is Eulerian if and only if every minimal cut has even cardinality. By Theorem 2.43, G is bipartite if G∗ is Eulerian, and G is Eulerian ifG∗ is bipartite. By Proposition 2.42, the converse is also true. 2 Anabstract dualofGis a graphGfor which there is a bijectionχ:E(G)→ E(G)such that F is the edge set of a circuit iff χ(F)is a minimal cut in G and vice versa. Theorem 2.43 shows that any planar dual is also an abstract dual.
The converse is not true. However, Whitney [1933] proved that a graph has an abstract dual if and only if it is planar (Exercise 34). We shall return to this duality relation when dealing with matroids in Section 13.3.
Exercises
1. LetG be a simple undirected graph onn vertices which is isomorphic to its complement. Show thatnmod 4∈ {0,1}.
2. Prove that every simple undirected graph G with |δ(v)| ≥ 12|V(G)| for all v∈V(G)is Hamiltonian.
Hint: Consider a longest path in Gand the neighbours of its endpoints.
(Dirac [1952])
3. Prove that any simple undirected graph G with|E(G)| >|V(G)|−1
2
is con-nected.
4. LetG be a simple undirected graph. Show that G or its complement is con-nected.
Exercises 43 5. Prove that every simple undirected graph with more than one vertex contains two vertices that have the same degree. Prove that every tree (except a single vertex) contains at least two leaves.
6. LetG be a connected undirected graph, and let (V(G),F)be a forest in G.
Prove that there is a spanning tree(V(G),T)with F⊆T ⊆E(G).
7. Let (V,F1) and (V,F2) be two forests with |F1| < |F2|. Prove that there exists an edgee∈ F2\F1 such that(V,F1∪ {e})is a forest.
8. Prove that any cut in an undirected graph is the disjoint union of minimal cuts.
9. LetGbe an undirected graph,C a circuit and Da cut. Show that|E(C)∩D|
is even.
10. Show that any undirected graph has a cut containing at least half of the edges.
11. Let(U,F)be a cross-free set system with|U| ≥2. Prove thatF contains at most 4|U| −4 distinct elements.
12. LetGbe a connected undirected graph. Show that there exists an orientation GofGand a spanning arborescenceT ofGsuch that the set of fundamental circuits with respect toT is precisely the set of directed circuits inG. Hint:Consider aDFS-tree.
(Camion [1968], Crestin [1969])
13. Describe a linear-time algorithm for the following problem: Given an adja-cency list of a graph G, compute an adjacency list of the maximal simple subgraph ofG. Do not assume that parallel edges appear consecutively in the input.
14. Given a graph G (directed or undirected), show that there is a linear-time algorithm to find a circuit or decide that none exists.
15. LetGbe a connected undirected graph,s∈V(G)andT aDFS-tree resulting from runningDFSon(G,s).s is called the root ofT.x is a predecessor of yinT ifx lies on the (unique)s-y-path in T.x is a direct predecessor of y if the edge{x,y}lies on the s-y-path inT.yis a (direct) successor of xif x is a (direct) predecessor of y. Note that with this definition each vertex is a successor (and a predecessor) of itself. Every vertex exceptshas exactly one direct predecessor. Prove:
(a) For any edge{v, w} ∈ E(G), v is a predecessor or a successor of w in T.
(b) A vertexv is an articulation vertex ofG if and only if – eitherv=sand|δT(v)|>1
– orv=s and there is a direct successorwof v such that no edge in G connects a proper predecessor of v (that is, excluding v) with a successor ofw.
∗ 16.Use Exercise 15 to design a linear-time algorithm which finds the blocks of an undirected graph. It will be useful to compute numbers
α(x):=min{f(w):w=xor {w,y} ∈E(G)\T for some successor yof x}
recursively during theDFS. Here (R,T)is theDFS-tree (with root s), and the f-values represent the order in which the vertices are added to R (see
theGraph Scanning Algorithm). If for some vertex x ∈ R\ {s}we have α(x)≥ f(w), where w is the direct predecessor ofx, thenwmust be either the root or an articulation vertex.
17. Prove:
(a) An undirected graph is 2-edge-connected if and only if it has at least two vertices and an ear-decomposition.
(b) A digraph is strongly connected if and only if it has an ear-decomposition.
(c) The edges of an undirected graph G with at least two vertices can be oriented such that the resulting digraph is strongly connected if and only if Gis 2-edge-connected.
(Robbins [1939])
18. A tournament is a digraph such that the underlying undirected graph is a (simple) complete graph. Prove that every tournament contains a Hamilto-nian path (R´edei [1934]). Prove that every strongly connected tournament is Hamiltonian (Camion [1959]).
19. Prove that if a connected undirected simple graph is Eulerian then its line graph is Hamiltonian. What about the converse?
20. Prove that any connected bipartite graph has a unique bipartition. Prove that any non-bipartite undirected graph contains an odd circuit as an induced sub-graph.
21. Prove that a strongly connected digraph whose underlying undirected graph is non-bipartite contains a (directed) circuit of odd length.
∗ 22.Let G be an undirected graph. A tree-decompositionof G is a pair (T, ϕ), whereT is a tree andϕ :V(T)→2V(G)satisfies the following conditions:
– for eache∈ E(G)there exists at∈V(T)withe⊆ϕ(t);
– for eachv∈V(G)the set{t ∈V(T):v∈ϕ(t)}is connected in T. We say that the width of(T, ϕ)is maxt∈V(T)|ϕ(t)| −1. Thetree-widthof a graphG is the minimum width of a tree-decomposition of G. This notion is due to Robertson and Seymour [1986].
Show that the graphs of tree-width at most 1 are the forests. Moreover, prove that the following statements are equivalent for an undirected graphG:
(a) G has tree-width at most 2;
(b) G does not contain K4 as a minor;
(c) G can be obtained from an empty graph by successively adding bridges and doubling and subdividing edges. (Doubling an edge e = {v, w} ∈ E(G) means adding another edge with endpointsv and w; subdividing an edge e= {v, w} ∈E(G)means adding a vertexx and replacing eby two edges {v,x},{x, w}.)
Note:Because of the construction in (c) such graphs are called series-parallel.
23. Show that if a graphGhas a planar embedding where the edges are embedded by arbitrary Jordan curves, then it also has a planar embedding with polygonal arcs only.
24. LetGbe a 2-connected graph with a planar embedding. Show that the set of circuits bounding the finite faces constitute a cycle basis ofG.
Exercises 45 25. Can you generalize Euler’s formula (Theorem 2.32) to disconnected graphs?
26. Show that there are exactly five Platonic graphs (corresponding to the Platonic solids; cf. Exercise 11 of Chapter 4), i.e. 3-connected planar regular graphs whose faces are all bounded by the same number of edges.
Hint:Use Euler’s formula (Theorem 2.32).
27. Deduce from the proof of Kuratowski’s Theorem 2.39:
(a) Every 3-connected simple planar graph has a planar embedding where each edge is embedded by a straight line and each face, except the outer face, is convex.
(b) There is a polynomial-time algorithm for checking whether a given graph is planar.
∗ 28.Given a graphGand an edgee= {v, w} ∈E(G), we say thatH results from G by subdividing e if V(H) = V(G) ∪ {x}. and E(H) = (E(G)\ {e})∪ {{v,x},{x, w}}. A graph resulting fromG by successively subdividing edges is called asubdivisionof G.
(a) Trivially, ifH contains a subdivision ofGthenGis a minor of H. Show that the converse is not true.
(b) Prove that a graph containing a K3,3 or K5 minor also contains a subdi-vision of K3,3or K5.
Hint:Consider what happens when contracting one edge.
(c) Conclude that a graph is planar if and only if no subgraph is a subdivision of K3,3 or K5.
(Kuratowski [1930])
29. Prove that each of the following statements implies the other:
(a) For every infinite sequence of graphs G1,G2, . . .there are two indices i < j such thatGi is a minor ofGj.
(b) LetG be a class of graphs such that for each G∈G and each minor H of G we have H ∈G (i.e.G is a hereditary graph property). Then there exists a finite set X of graphs such that G consists of all graphs that do not contain any element ofX as a minor.
Note:The statements have been proved by Robertson and Seymour; they are a main result of their series of papers on graph minors (not yet completely published). Theorem 2.39 and Exercise 22 give examples of forbidden minor characterizations as in (b).
30. LetG be a planar graph with an embedding , and let C be a circuit of G bounding some face of . Prove that then there is an embedding of G such thatC bounds the outer face.
31. (a) LetG be disconnected with an arbitrary planar embedding, and let G∗ be the planar dual with a standard embedding. Prove that (G∗)∗ arises fromGby successively applying the following operation, until the graph is connected: Choose two vertices x and y which belong to different connected components and which are adjacent to the same face; contract {x,y}.
(b) Generalize Corollary 2.45 to arbitrary planar graphs.
Hint:Use (a) and Theorem 2.26.
32. LetG be a connected digraph with a fixed planar embedding, and letG∗ be the planar dual with a standard embedding. How areG and(G∗)∗ related?
33. Prove that if a planar digraph is acyclic (strongly connected), then its planar dual is strongly connected (acyclic). What about the converse?
34. (a) Show that ifG has an abstract dual and H is a minor ofG then H also has an abstract dual.
∗ (b) Show that neither K5 norK3,3 has an abstract dual.
(c) Conclude that a graph is planar if and only if it has an abstract dual.
(Whitney [1933])
References
General Literature:
Berge, C. [1985]: Graphs. 2nd revised edition. Elsevier, Amsterdam 1985 Bollob´as, B. [1998]: Modern Graph Theory. Springer, New York 1998
Bondy, J.A. [1995]: Basic graph theory: paths and circuits. In: Handbook of Combinatorics;
Vol. 1 (R.L. Graham, M. Gr¨otschel, L. Lov´asz, eds.), Elsevier, Amsterdam 1995 Bondy, J.A., and Murty, U.S.R. [1976]: Graph Theory with Applications. MacMillan,
Lon-don 1976
Diestel, R. [1997]: Graph Theory. Springer, New York 1997
Wilson, R.J. [1972]: Introduction to Graph Theory. Oliver and Boyd, Edinburgh 1972 (3rd edition: Longman, Harlow 1985)
Cited References:
Aoshima, K., and Iri, M. [1977]: Comments on F. Hadlock’s paper: finding a maximum cut of a Planar graph in polynomial time. SIAM Journal on Computing 6 (1977), 86–87 Camion, P. [1959]: Chemins et circuits hamiltoniens des graphes complets. Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences (Paris) 249 (1959), 2151–2152 Camion, P. [1968]: Modulaires unimodulaires. Journal of Combinatorial Theory A 4 (1968),
301–362
Dirac, G.A. [1952]: Some theorems on abstract graphs. Proceedings of the London Math-ematical Society 2 (1952), 69–81
Edmonds, J., and Giles, R. [1977]: A min-max relation for submodular functions on graphs.
In: Studies in Integer Programming; Annals of Discrete Mathematics 1 (P.L. Hammer, E.L. Johnson, B.H. Korte, G.L. Nemhauser, eds.), North-Holland, Amsterdam 1977, pp.
185–204
Euler, L. [1736]: Solutio Problematis ad Geometriam Situs Pertinentis. Commentarii Academiae Petropolitanae 8 (1736), 128–140
Euler, L. [1758]: Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Petropolitanae 4 (1758), 140–
160
Hierholzer, C. [1873]: ¨Uber die M¨oglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen 6 (1873), 30–32
Hopcroft, J.E., and Tarjan, R.E. [1974]: Efficient planarity testing. Journal of the ACM 21 (1974), 549–568
Kahn, A.B. [1962]: Topological sorting of large networks. Communications of the ACM 5 (1962), 558–562
References 47 Knuth, D.E. [1968]: The Art of Computer Programming; Vol. 1; Fundamental Algorithms.
Addison-Wesley, Reading 1968 (third edition: 1997)
K¨onig, D. [1916]: ¨Uber Graphen und Ihre Anwendung auf Determinantentheorie und Men-genlehre. Mathematische Annalen 77 (1916), 453–465
K¨onig, D. [1936]: Theorie der endlichen und unendlichen Graphen. Chelsea Publishing Co., Leipzig 1936, reprint New York 1950
Kuratowski, K. [1930]: Sur le probl`eme des courbes gauches en topologie. Fundamenta Mathematicae 15 (1930), 271–283
Legendre, A.M. [1794]: ´El´ements de G´eom´etrie. Firmin Didot, Paris 1794
Minty, G.J. [1960]: Monotone networks. Proceedings of the Royal Society of London A 257 (1960), 194–212
Moore, E.F. [1959]: The shortest path through a maze. Proceedings of the International Symposium on the Theory of Switching; Part II. Harvard University Press 1959, pp.
285–292
R´edei, L. [1934]: Ein kombinatorischer Satz. Acta Litt. Szeged 7 (1934), 39–43
Robbins, H.E. [1939]: A theorem on graphs with an application to a problem of traffic control. American Mathematical Monthly 46 (1939), 281–283
Robertson, N., and Seymour, P.D. [1986]: Graph minors II: algorithmic aspects of tree-width. Journal of Algorithms 7 (1986), 309–322
Tarjan, R.E. [1972]: Depth first search and linear graph algorithms. SIAM Journal on Computing 1 (1972), 146–160
Thomassen, C. [1980]: Planarity and duality of finite and infinite graphs. Journal of Com-binatorial Theory B 29 (1980), 244–271
Thomassen, C. [1981]: Kuratowski’s theorem. Journal of Graph Theory 5 (1981), 225–241 Tutte, W.T. [1961]: A theory of 3-connected graphs. Konink. Nederl. Akad. Wetensch.
Proc. A 64 (1961), 441–455
Wagner, K. [1937]: ¨Uber eine Eigenschaft der ebenen Komplexe. Mathematische Annalen 114 (1937), 570–590
Whitney, H. [1932]: Non-separable and planar graphs. Transactions of the American Math-ematical Society 34 (1932), 339–362
Whitney, H. [1933]: Planar graphs. Fundamenta Mathematicae 21 (1933), 73–84
In this chapter we review the most important facts about Linear Programming.
Although this chapter is self-contained, it cannot be considered to be a compre-hensive treatment of the field. The reader unfamiliar with Linear Programming is referred to the textbooks mentioned at the end of this chapter.
The general problem reads as follows: