Multigraphs and Digraphs - BUKU GRAPHS & DIGRAPH PDF

There are occasions when a graph is not the appropriate structure to model a particular situation. For example, suppose that we are considering various locations in a certain community and there are roads between some pairs of locations that do not pass through any other location. Although this situation may be represented by a graph, there may be some characteristics in this net- work of roads that are not captured by a graph. For example, suppose that there are pairs of locations connected by two or more roads (not passing through any other location) and this information is important to us. Or perhaps the roads between locations are all one-way streets and this too is important to us. This leads us to different structures that are similar to but not identical to graphs.

Multigraphs

In the definition of a graphG, every two distinct vertices are joined by either one edge or no edge ofG. There will be occasions when we will want to permit more than one edge to join two vertices. A multigraph is a nonempty set of vertices, every two of which are joined by a finite number of edges. Hence a multigraph H may be expressed as H = (V, E), where E is a multiset of 2-element subsets ofV. Two or more edges that join the same pair of distinct vertices are calledparallel edges. Theunderlying graphof a multigraphH is that graphGfor whichV(G) =V(H) anduv∈E(G) ifuandv are joined by at least one edge inH.

An edge joining a vertex to itself is called aloop. Structures that permit both parallel edges and loops (including parallel loops) are often calledpseudo- graphs. For emphasis then, every two vertices of a graph are joined by at most

46 CHAPTER 1. INTRODUCTION TO GRAPHS one edge and loops are not permitted. In a multigraph, every two vertices are permitted to be joined by more than one edge but this is not required. Also, no multigraph contains a loop. In a pseudograph, every two vertices are permitted to be joined by more than one edge and loops are permitted. However, parallel edges and loops are not required in pseudographs. There are authors who refer to multigraphs or pseudographs as graphs and those who refer to what we call graphs assimple graphs. Consequently, when reading any material written on graph theory, it is essential that there is a clear understanding of how the termgraph is being used. According to the terminology introduced here then, every multigraph is a pseudograph and every graph is both a multigraph and a pseudograph.

In Figure 1.45,H₁ and H₄ are multigraphs while H₂ and H₃ are pseudographs. Of course,H₁andH₄are also pseudographs whileH₄is the only graph in Figure 1.45. For a vertex v in a multigraphG, the degreedegv ofv in G is the number of edges of Gincident with v. In a pseudograph, each loop at a vertex contributes 2 to its degree. For the pseudograph H₃ of Figure 1.45, degu= 5 and degv= 2.

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Figure 1.45: Multigraphs and pseudographs

When describing walks in multigraphs or in pseudographs, it is often nec- essary to list edges in the sequence as well as vertices in order to specify the edges being used in the walk. For example,

W = (u, e₁, u, v, e₆, w, e₆, v, e₇, w) is au−wwalk in the pseudographGof Figure 1.46.

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Figure 1.46: Walks in a pseudograph

Digraphs

Adirected graphordigraphD is a finite nonempty set of objects called vertices together with a (possibly empty) set of ordered pairs of distinct vertices of D called arcs or directed edges. As with graphs, the vertex set of D is denoted by V(D) and the arc set (or directed edge set) of D is denoted by E(D). A digraph D with vertex set V = {u, v, w, x} and arc set E ={(u, v),(v, u),(u, w),(w, v),(w, x)} is shown in Figure 1.47. Observe that when a digraph is described by means of a diagram, the “direction” of each arc is indicated by an arrowhead.

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Figure 1.47: A digraph

Much of the terminology used for digraphs is quite similar to that used for graphs. The cardinality of the vertex set of a digraphD is called theorderof D and is ordinarily denoted byn, while the cardinality of its arc set is thesize ofDand is ordinarily denoted bym. Ifa= (u, v) is an arc of a digraphD, then a is said to joinu andv. The vertexuis said to be adjacent to v and v is adjacent fromu. For a vertexvin a digraphD, theoutdegreeodvofvis the number of vertices ofDto whichv is adjacent, while the indegreeidvofv is the number of vertices ofDfrom whichvis adjacent. Theout-neighborhood N⁺(v) of a vertexv in a digraphD is the set of vertices adjacent fromv, while the in-neighborhood N⁻(v) of v is the set of vertices adjacent to v. Thus odv=|N⁺(v)| and idv=|N⁻(v)|. The degreedegv of a vertexv is defined by

degv = odv+ idv.

For the vertexvin the digraph of Figure 1.48, odv= 3, idv= 2 and degv= 5.

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Figure 1.48: The outdegree, indegree and degree of a vertex The directed graph version of Theorem 1.4 is stated next.

48 CHAPTER 1. INTRODUCTION TO GRAPHS Theorem 1.23 (The First Theorem of Digraph Theory) If D is a digraph of size m, then

v∈V(G)

odv= X

v∈V(G)

idv=m.

Proof. When the outdegrees of the vertices are summed, each arc is counted once. Similarly, when the indegrees of the vertices are summed, each arc is counted just once.

A digraphD₁ is isomorphic to a digraph D₂, writtenD₁ ∼

=D₂, if there exists a bijective functionφ:V(D₁)→V(D₂) such that (u, v)∈E(D₁) if and only if (φ(u), φ(v))∈E(D₂). The functionφ is called anisomorphism from D₁ toD₂.

There is only one digraph of order 1, namely the trivial digraph. Also, there is only one digraph of order 2 and sizemfor eachmwith 0≤m≤2. There are four digraphs of order 3 and size 3, all of which are shown in Figure 1.49.

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Figure 1.49: The digraphs of order 3 and size 3

A digraph D₁ is a subdigraph of a digraph D if V(D₁) ⊆ V(D) and E(D₁) ⊆ E(D). We use D₁ ⊆ D to indicate that D₁ is a subdigraph of D.

A subdigraph D₁ of D is a spanning subdigraph of D if V(D₁) = V(D).

Vertex-deleted, arc-deleted, induced and arc-induced subdigraphs are defined in the expected manner. These last two concepts are illustrated for the digraph D of Figure 1.50, where

V(D) ={v₁, v₂, v₃, v₄},U ={v₁, v₂, v₃}, andX ={(v₁, v₂),(v₂, v₄)}.

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D v₄

D[U]

v₄ D[X] Figure 1.50: Induced and arc-induced subdigraphs

We now consider certain types of digraphs that occur periodically. A digraph is symmetricif whenever (u, v) is an arc ofD, then (v, u) is an arc ofD as well. There is a natural one-to-one corresponding between the set of symmetric digraphs and the set of graphs. The complete symmetric digraph Kn^∗ of order nhas both arcs (u, v) and (v, u) for every two distinct verticesuandv.

A digraph is called anoriented graphif whenever (u, v) is an arc ofD, then (v, u) is not an arc ofD. Thus an oriented graphD can be obtained from a graphGby assigning a direction to (or by “orienting”) each edge ofG, thereby transforming each edge of a graphGinto an arc and transformingGitself into an oriented graph. The digraphD is also called anorientationofG. Figure 1.51 shows three digraphs D₁, D₂ and D₃. While D₁ is a symmetric digraph and D₂ is an oriented graph, the digraphD₃is neither. Theunderlying graphof a digraph D is that graph obtained by replacing each arc (u, v) or symmetric pair (u, v), (v, u) of arcs by the edgeuv. The underlying graph of each digraph in Figure 1.51 is the graphG.

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Figure 1.51: Digraphs with the same underlying graph

An orientation of a complete graph is called a tournament and will be studied in some detail in Chapter 4. A digraph D is regular of degree r or r-regular if odv = idv =r for every vertex v of D. A 1-regular digraph D₁ and a 2-regular digraph D₂ are shown in Figure 1.52. The digraphD₂ is a tournament.

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Figure 1.52: Regular digraphs

The terms walk, open and closed walk, trail, path, circuit and cycle for graphs have natural counterparts in digraph theory as well, the important dif- ference being that the directions of the arcs must be followed in each of these

50 CHAPTER 1. INTRODUCTION TO GRAPHS walks. In particular, when referring to digraphs, the terms directed path, directed cycle and directed circuitare synonymous with the terms path, cycle and circuit. More formally, for vertices u and v in a digraph D, a directed u−v walk(or simply a u−v walk) inD is a finite sequence

(u=u₀, u₁, u₂, . . . , uk=v)

of vertices, beginning with uand ending with v such that (uⁱ, ui+1) is an arc for 0≤i≤k−1. The numberkof occurrences of arcs (including repetition) in the walk is its length. Digraphs in which every vertex has positive outdegree must contain cycles (see Exercise 14).

Theorem 1.24 If Dis a digraph such thatodv≥k≥1for every vertexv of D, thenD contains a cycle of length at least k+ 1.

Connected Digraphs

A digraphDisconnected(orweakly connected) if the underlying graph ofDis connected. A digraphDisstrong(orstrongly connected) if for every pairu, vof vertices,D contains both au−vpath and a v−upath. While all digraphs of Figure 1.53 are connected, onlyD₁is strong.

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Figure 1.53: Connectedness properties of digraphs

Distance can be defined in digraphs as well. For verticesuandvin a digraph D containing au−v path, the (directed)distance d(u, v) from~ uto vis the length of a shortestu−v path inD. Thus the distancesd(u, v) and~ d(v, u) are~ defined for all pairs u, v of vertices in a digraphD if and only ifD is strong.

This distance is not a metric, in general. Although directed distance satisfies the triangle inequality, it is not symmetric unlessDis symmetric, in which case D can be considered a graph. Eccentricity can be defined as before, as well as radius and diameter in a strong digraph D. The eccentricity e(u) of a vertexuinDis the distance fromuto a vertex farthest fromu. The minimum eccentricity of the vertices ofDis theradiusrad(D) ofD, while thediameter diam(D) is the greatest eccentricity.

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Figure 1.54: Eccentricities in a strong digraph

Each vertex of the strong digraphDof Figure 1.54 is labeled with its eccentricity. Observe that rad(D) = 2 and diam(D) = 5, so it is not true, in general, that diam(D)≤2 rad(D), as is the case with graphs.

As with multigraphs or pseudographs, if more than one arc in the same direction is permitted to join two vertices in a digraph, then these arcs are parallel arcs and a multidigraph results. A directed loop in a digraph is an arc that joins a vertex to itself. In apseudodigraph, parallel arcs and directed loops (including parallel directed loops) are permitted. A multidigraph D₁ and a pseudodigraphD₂ are shown in Figure 1.55.

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D₂: D₁:

Figure 1.55: A multidigraph and a pseudodigraph

Exercises for Section 1.4

1. Show that for every connected graphGof order 3 or more, there exists a multigraphH whose underlying graph isGand where distinct vertices of H have distinct degrees.

2. Prove for every connected graph Gof order n= 3 or n= 4 and size m that it is possible to label the edges ofGbye₁, e₂, . . . , emand replaceei

by i parallel edges for each i (1 ≤i ≤ m) such that the degrees of the vertices of the resulting multigraph H are distinct.

3. Determine which of the following sequences are the degree sequences of a multigraph.

52 CHAPTER 1. INTRODUCTION TO GRAPHS (a) s₁: 3,2,1 (b)s₂: 5,2,1

(c)s₃: 6,4,2 (d)s₄: 3,2,2 (e)s₅: 4,3,2,1 (f)s₆: 5,3,2,1 (g)s₇: 4,4,4,4 (h)s₈: 7,5,3,1.

4. Prove that a sequence s : d₁, d₂, . . . , dn (n ≥ 1) of nonnegative integers with d₁ ≥ d₂ ≥ · · · ≥ dn is the degree sequence of a multigraph if and only ifPⁿ

i=1di is even andd₁≤¹

Pⁿ

i=1di.

5. LetGbe a connected graph of ordernwhere the vertices ofGare labeled as v₁, v₂, . . . , vn in some way. A multigraph H of size m with V(H) = V(G) is obtained by replacing each edge vivj of Gby min{i, j} parallel edges.

(a) FindmifG=K₅.

(b) Find sharp upper and lower bounds form ifG=C₅. (c) Find the minimum value ofmifGis bipartite.

6. Determine all digraphs of order 4 and size 4.

7. Prove or disprove: For every integer n≥2, there exists a digraphD of ordernsuch that for every two distinct verticesuandvofD, odu6= odv and idu6= idv.

8. Prove or disprove: There exists a nontrivial digraph D in which no two vertices ofD have the same outdegree but every two vertices ofD have the same indegree.

9. Prove or disprove: No digraph contains an odd number of vertices of odd outdegree or an odd number of vertices of odd indegree.

10. Prove or disprove: IfD₁ andD₂ are two digraphs withV(D₁) ={u₁, u₂, . . ., un} and V(D₂) ={v₁, v₂, . . ., vn} such that id^D1ui = id^D2vi and od^D1ui= od^D2vi fori= 1,2, . . . , n, thenD₁∼

=D₂.

11. Prove that there exist regular tournaments of every odd order but there are no regular tournaments of even order.

12. LetT be a tournament withV(T) ={v₁, v₂,. . .,vn}. We know that

i=1

odvi=

i=1

idvi= n

(a) Prove thatPⁿ

i=1(odvi)²=Pⁿ

i=1(idvi)². (b) Prove or disprove: Pⁿ

i=1(odvi)³=Pⁿ

i=1(idvi)³.

13. Theadjacency matrix A(D) of a digraphDwithV(D) ={v₁,v₂,. . ., vn} is the n×n matrix [a^ij] defined by aij = 1 if (vⁱ, vj) ∈ E(D) and aij = 0 otherwise.

(a) What information do the row sums and column sums of the adjacency matrix of a digraph provide?

(b) Characterize matrices that are adjacency matrices of digraphs.

14. (a) Prove Theorem 1.24: If D is a digraph such that odv ≥k ≥1 for every vertexvofD, thenD contains a cycle of length at leastk+ 1.

(b) Prove that ifD is a digraph such that idv≥k≥1 for every vertex v ofD, thenD contains a cycle of length at leastk+ 1.

15. Let G be a connected graph of order n ≥ 3. Prove that there is an orientation ofG in which no directed path has length 2 if and only ifG is bipartite.

16. Prove that for every two positive integersaandbwitha≤b, there exists a strong digraph Dwith rad(D) =aand diam(D) =b.

17. Thecenter Cen(D) of a strong digraphD is the subdigraph induced by those vertices v withe(v) = rad(D). Prove that for every oriented graph D₁, there exists a strong oriented graphDsuch that Cen(D) =D₁. 18. Prove that every digraphDcontains a setSof vertices with the properties

(1) no two vertices inSare adjacent inD and (2) for every vertexv ofD not inS, there exists a vertexuin S such thatd~(u, v)≤2.

19. Let S be a finite nonempty set of positive integers. The divisor pseudodigraph D determined by S hasS as its vertex set and for two (not necessarily distinct) verticesuandv, (u, v) is a directed edge ofDifu|v.

The divisor graph G determined by S hasS as its vertex set and uv (u6=v) is an edge ofGif eitheru|v orv|u.

(a) Determine the divisor pseudodigraph forS={1,2,3,6}. (b) Prove or disprove: There exists a divisor graphGwithG∼

=K₅. (c) Prove or disprove: There exists a divisor graphGwithG∼

=P₅. (d) Prove or disprove: There exists a divisor graphGwithG∼

=C₅. (e) Prove or disprove: There exists a divisor graphGwithG∼=C₆. (f) Prove or disprove: There exists a divisor graphGwithG∼=C₇. (g) Prove or disprove: There exists a divisor graphGwithG∼=C₇.

Chapter 2 Trees and Connectivity

In nearly every concept, problem and theorem that we encounter, we are pri- marily concerned with connected graphs. There are various ways of measuring connectedness for graphs and in many instances, it will be important for us to know the degree of connectedness of the graphs being considered. While a graph may contain au−vpath for every pairu, vof vertices and consequently is connected, a graph can satisfy this requirement in minimal ways or in much stronger ways.

Dalam dokumen BUKU GRAPHS & DIGRAPH PDF (Halaman 58-68)