Conclusiones

Definition 7.6. Agraphis a pairG= (V,E)of sets such that E ⊆[V]2. The elements ofV are theverticesof the graphG; the elements ofE are itsedges. Figure 7.3 shows a drawing of

7. Algorithmic Design and Theory

a graph. How the vertices and edges are drawn is considered irrelevant; all that matters is the information of which pairs of vertices form an edge and which do not.

1 4

2

3 5 7

6

Figure 7.3:The graph onV={1,...,7}with edge setE={1,2},{1,5},{2,5},{3,4},{5,7}.

Definition 7.7. LetG= (V,E)be a graph. Two verticesx,yofGareneighboursif{x,y}is an edge ofG. Two edgese=f areadjacentif they share a vertex. The set of neighbours of a vertex vinGis denoted byN(v). Thedegree d(v)of a vertexvis the number of neighbours ofv.

Definition 7.8. LetG= (V,E)andG= (V,E)be two graphs. IfV⊆V andE⊆E, thenG is asubgraphofG, written asG⊆G. IfG⊆GandG=G, thenGis aproper subgraphofG. IfG⊆GandGcontains all the edgesxy∈E withx,y∈V, thenGis aninduced subgraphof Gand written asG:=G[V](figure 7.4).

G G G

Figure 7.4:A graphGwith subgraphsGandG:Gis an induced subgraph ofG, butGis not.

Definition 7.9. Apathis a non-empty graphP= (V,E)of the form V={x0,x1,...,xk} E={x0x1,x1x2,...,xk−1xk},

where thexiare all distinct. The number of edges of a path is itslength, and the path of lengthk

is denoted byPk(figure 7.5). A path is often referred to by the natural sequence of its vertices:

7.1. Mathematical Concepts

P=x0x1...xk. For 0≤i≤ j≤ksubpaths ofPcan be written as

Pxi:=x0...xi

xiP:=xi...xk

xiPxj:=xi...xj

G P

Figure 7.5:A pathP=P6inG.

Definition 7.10. LetP=x0x1...xk−1be a path. Then ifk≥3, the graphC:=P+xk−1x0is called

acycle. As with paths, a cycle is often denoted by its (cyclic) sequence of vertices;x0...xk−1x0.

Thelength of a cycle is its number of edges and is denoted byCk_{. An edge which joins two}

vertices of a cycle but is not itself an edge of the cycle is achordof that cycle. Aninduced cycle inG, a cycle inGforming an induced subgraph, is one that has no chords (figure 7.6).

y x

Figure 7.6:A cycleC8_{with chordxy, and induced cyclesC}5_,C4_.

Definition 7.11. A graphG= (V,E)is calledconnected if it is non-empty and any two of its vertices are linked by a path inG. Amaximal connected subgraphofG,G= (V,E), is a graph which is connected and for all verticesusuch thatu∈V andu∈/V, there is no vertexv∈V for whichuv∈E. A maximal connected subgraph ofGis acomponentofG(figure 7.7).

Definition 7.12. Anacylicgraph, one not containing any cycles, is called aforest. A connected forest is called atree. Thus a forest is a graph whose components are trees. The vertices of degree 1 in a tree are itsleaves. Sometimes it is convenient to consider one vertex of a tree as

7. Algorithmic Design and Theory

Figure 7.7:A three component graph

special; such a vertex is then called therootof this tree. The root of a tree is never called a leaf, even if it has degree 1. A treeTwith a fixed rootris arooted tree(figure 7.8).

r

Figure 7.8:A tree with rootr.

Definition 7.13. Adirected graphis a pair(V,E)of disjoint sets (of vertices and edges) together with two maps init :E →V and ter :E→V assigning to every edgeeaninitial vertexinit(e) and aterminal vertexter(e). The edgeeis said to bedirected frominit(e)to ter(e). A directed graph may have several edges between the same two vertices x,y called multiple edges. If init(e) =ter(e), the edgeeis called aloop. Anoriented graphis a directed graph without loops or multiple edges.

Definition 7.14. Avertex colouringof a graphG= (V,E)is a mapc:V →Ssuch thatc(v)= c(w)whenevervandware adjacent. The elements of the setSare called the availablecolours. Anedge colouringofG is a mapc:E →Swithc(e)=c(f)for any adjacent edgese,f. A weakening of the vertex and edge colouring definitions by allowingc(v) =c(w)andc(e) =c(f) respectively leads to themolecular graph. A molecular graph M= (A,B)has a weak vertex colouringcv:A→PwherePis the set of atomic numbers for elements in the periodic table, and

a weak edge colouringce:B→RwhereRis the set of allowable bond orders. This definition

comes from Nicet al.38

Definition 7.15. Aplane graphis a pair(V,E)of finite sets with the following properties: