Graphs and Networks

One of the most elementary discrete structures to model networking problems are undirected and directed graphs.

1.2.2.1 Undirected Graphs

An undirected graph, or short graph, is a pair G= (V,E)consisting of a set of vertices V and a set of edges E where each edge e∈E is a two-element unordered subset of V . Hence, we also write{i,j} ∈E with i,j∈V . We further say that an edge{i,j} ∈E is incident to both i and j. Figures 1(a) and 1(b) show two famous graphs, the cycle on five vertices (denoted by C₅) and the Petersen graph.

Undirected graphs are used to model relations between entities that do not have a direction associated with them or where the direction does not play a role. In com-munication networks, undirected graphs are used to describe, for example, the topol-ogy of an optical fiber network, where the nodes represent the routers and an edge exists in the graph if and only if there is a direct optical fiber connection (link) be-tween the routers (Figure 1(c) shows the Pan-European Triangular Topology graph defined by COST action 266). Another example is the modeling of potential con-flicts between access points of a Wireless Local Access Network (WLAN). Here the

(a) Graph C5 (b) Petersen graph (c) COST 266 topology graph

Fig. 1.1 Three undirected graphs: A cycle with five vertices and five edges, the Petersen graph with ten vertices and 15 edges, and the triangular COST 266 topology graph with 28 vertices and 61 edges

nodes represent the access points and two nodes are adjacent if and only if there are locations in the designated coverage area where the signals would interfere if both access points would be assigned the same radio frequency (see Chapter 11 for more information).

If not stated otherwise we assume that a graph is simple in the sense that there are no parallel edges (identical elements of E) or loops (edges of the form{i,i}).

Two distinct vertices i,j∈V are called adjacent or neighbors if{i,j} ∈E. This concept is extended to subsets of vertices by the function N : 2^V→2^V that assigns to every subset S⊆V all neighboring vertices that are not part of the subset, i.e., N(S) ={j∈V\S| {i,j} ∈E,i∈S}. If S={i} we simplify notation by writing N(i)instead of N({i}). Similarly, the functionδ : 2^V →2^E assigns to every vertex subset S the edges that connect S with V\S. The degree of a vertex is defined by the function deg : V→Z+which assigns to a vertex i∈V the number of adjacent edges, deg(i) =|δ(i)|. If the graph G might not be clear from the context, a subscript such as degGis used for all three functions.

Given a graph G= (V,E), we define the complement of G as ¯G= (V,E)¯ with E¯={{i,j} | {i,j} ∈E}.

A graph G= (V,E)is called bipartite if the vertex set can be partitioned into two subsets V1,V₂such that for every edge{i,j} ∈E, i∈V1and j∈V2. In other words, for all i∈V1, N(i)⊆V2and for all i∈V2, N(i)⊆V1. Bipartite graphs are therefore sometimes denoted by(V1,V2,E).

A graph G= (V,E)is called complete if all vertices are mutually adjacent, i.e., {i,j} ∈E for all i,j∈V , i=j.

A graph H= (U,F)is called a subgraph of G= (V,E)if U ⊆V and F⊆E.

If F={{i,j} ∈E|i,j∈U}, G[U] =H is the subgraph of G induced by U . The subgraph of the Petersen graph induced by{1,2,3,4,5}is C5.

A path p in a graph G is a sequence(i0,e1,i1,e2,i2, . . . ,i_k−1,ek,ik)of k+1 ver-tices and k edges (k≥1) with the property that ej={ij−1,ij}. We write e∈p and

i∈p for edges and vertices that are part of the path. A path is called simple if no vertex appears more than once in the sequence (and hence no edge appears more than once as well). A graph G is called connected if there exists a path between ev-ery pair of vertices. A component S⊆V of a graph G= (V,E)is a subset of vertices that induces a maximally connected subgraph G[S].

Two vertices i,j∈V are said to be k-edge-connected if there exist k edge-disjoint paths between i and j in G. The edge connectivity k(G)of a graph is the minimum over all vertex pairs of the number of edge-disjoint paths.

Similarly, two vertices i,j∈V are said to be k-vertex-connected if there exist k vertex-disjoint paths between i and j (except for vertices i and j). The vertex connectivity(G)of a graph is the minimum over all vertex pairs of the number of vertex-disjoint paths.

A circuit in a graph G= (V,E)is a closed path(i0,e1,i1,e2,i2, . . . ,i_k−1,ek,ik, i.e., i0=ik. A cycle is a circuit with the additional property that all vertices (and edges) except the start and end vertex are distinct.

A tree in a graph G= (V,E)is a cycle-free connected subgraphT = (I,L)with I⊆V and L⊆E. Hence, there exists a unique path between every pair of nodes i,j∈I. Note that in a cycle-free connected graph|L|=|I|−1. If I=V , thenT is a spanning tree.

A stable set or independent set S is a subset of the vertices such that no two vertices have an edge in common, i.e., if i,j∈S, then{i,j} ∈E. Stated otherwise, all vertices in the graph induced by S have degree zero. For the Petersen graph S={1,3,7}is a stable set. Since this set cannot be extended further without losing its stability, S is a maximal stable set. A maximum stable set is a stable set that is maximal and no other stable set has a higher cardinality, e.g., {1,3,9,10} is a maximum stable set.

A clique in a graph G= (V,E) is a subset S of the vertices such that G[S] is complete. Note that S is a clique in G if and only if S is a stable set in ¯G.

A matching M is a subset of the edges such that no two edges have a vertex in common, i.e., if e,f ∈M, then e∩f =/0 (note that edges are sets of two elements).

Stated otherwise, all vertices in the subgraph(V,M)have degree at most 1. For the Petersen graph, a (maximum) matching is given by{{1,2},{3,4},{6,8},{7,9}}.

For an arbitrary parameter bi∈Qfor all i∈V , we define the cumulative weight function b : 2^V →Qas b(S) =∑_i∈Sbi. Similarly, for an arbitrary parameter ce∈Q defined for all e∈E, we define the cumulative weight function c : 2^E→Qas c(L) =

∑e∈Lce.

1.2.2.2 Directed Graphs

A directed graph, or digraph, is a pair D= (V,A)consisting of a set of vertices V and a set of arcs A where each arc a∈A is a two-element ordered subset of V . Hence, we also write (i,j)∈A. Digraphs are used in those situations where the direction of the relation is of importance, for example, in communication networks in the modeling of a traffic flow from a source node to a sink node, where it is of

D

A B C

F E

Fig. 1.2 A directed graph with seven vertices and 12 arcs

importance to know in which direction the signal is transported between the routers.

Another example is the connection of wireless devices with variable transmission power. Here, an arc(i,j)exists if and only if j represents a device that is within the transmission reach of device i. Since each device has its own power control, device i is not automatically in reach of device j if j is in reach of i (see Chapter 13 for more on this application).

For digraphs, we distinguish between arcs coming in to a vertex i∈V and arcs going out i. The functions N⁻: V →2^V,δ⁻: V→2^A, deg⁻: V →Z+ (or N⁺: V →2^V,δ⁺: V→2^A, deg⁺: V→Z+) associate with every vertex i∈V the set of incoming neighbors, arcs, and degree (or outgoing neighbors, arcs, and degree).

A (directed) path p in a digraph D is a sequence(i0,a1,i1,a2,i2, . . . ,i_k−1,ak,ik) of k+1 vertices and k arcs (k≥1) with the property that aj= (i_j−1,ij). We denote a∈p if an arc a∈A is part of the path; similarly, i∈p. Again, a path is called simple if vertices are not repeated in the sequence.

A digraph is called strongly connected if there exists a path from any vertex to any other vertex. A (directed) cycle is a simple directed path with i0=i_k. A digraph is called a directed acyclic graph or DAG if it does not contain directed cycles.

An arborescence is a digraph with the property that there is a vertex v∈V such that there is exactly one directed path from v to every other vertex u∈V . The vertex v is called the root of the arborescence. Stated otherwise, an arborescence is a directed rooted tree with all arcs directed away from the root.