Assignment 2 - Slides for Combinatorial Computing | MAT 3770, Study notes of Mathematics

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Section 1.2. Isomorphism

I (^) Two graphs G = (VG ; EG ) and H = (VH ; EH ) are said to be isomorphic if there exists a bijection f : VG → VH 3 < u; w >∈ EG IFF < f (u); f (w) >∈ EH I (^) I.e., we can relabel the vertices of G to be vertices of H , maintaining the corresponding edges in G and H ; pairs are adjacent in G IFF pairs are adjacent in H 1

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I (^) The mapping from VG to VH given by f(1) = u, f(2) = v, f(3) = w, f(4) = x, f(5) = y, f(6) = z is the requisite bijection.

Isomorphic Graphs

I (^) Same number of vertices

I (^) Same number of edges

I (^) same number of vertices with a given degree

I (^) corresponding edges are maintained between vertices of same degree as preñimage.

A graph H = (VH ; EH ) is a subgraph of G = (VG ; EG ) if VH ⊆ VG and EH ⊆ EG.

Isomorphic Subgraphs

I (^) If we cannot nd isomorphic subgraphs, then the graphs are not isomorphic.

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a (^) d 1 4

deg 2: b,d,f,h 3,4,8,

deg 3: a,c,e,g 1,2,5,

I (^) Subgraphs containing these (deg 2) vertices must be isomorphic.

I No edges between b,d,f, or h (within same set), while

edges <3,4> and <7,8> exist. Therefore the two graphs

are not isomorphic.

I (^) Independent Set : Set of vertices no two of which share an edge

I (^) Maximal Independent Set : Cannot add any other vertex in the graph and remain independent (i.e., every vertex not in the set is adjacent to some vertex in the set)

I (^) Maximum Independent Set : Maximum cardinality of all Independent sets

I (^) Theorem : Given a graph G=(V, E), if S⊂V is independent, then vS is an edge cover and vice versa

Applications

Connected Components

I (^) One of the incentives for developing the Internet was the threat of war and the fear of having communications between various installations in the United States severed.

I (^) Given a graph, can we determine if there is a critical edge, one whose removal disconnects the graph?

Applications

Edge Cover

The Manhattan Police Department (MPD) knows several heads of organized crime are meeting in a particular area of the city and want to keep the streets there under surveillance. Unfortunately, owing to budget constraints, they need to use the fewest ofcers possible.

a b^ c

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How can we determine on which corners to place ofcers to maximize their usefulness (the number of adjacent blocks they can observe) while minimizing the number of ofcers?

Applications

Scheduling Problems

Suppose we allowed students to sign up for courses, then scheduled the courses so the total number of hours needed is minimized, and no two classes which share students meet at the same time.

This can be modeled with a graph where each class is a vertex, and an edge between two vertices means they share at least one student.

A set of courses can all meet at the same time if there are no edges between any of them, i.e., they form an independent set. Thus, we need to nd the minimum number of independent sets that collectively include all vertices.

Consider also that if we nd a maximum independent set, we'll have a minimum edge cover, and vice versa. Thus, nding a maximum independent set is equivalent to nding a minimum edge cover.

Applications

Not everyone has a telephone, nor even a permanent address. Suppose we needed to get word out ìon the streetî about a new program to help the homeless.

We can use a graph to represent this problem. Vertices are people, and the directed edges between them represent ìis able to contact.î

Because we have so little time, we want to nd a minimal subset of people who can spread the word to the whole groupóeither directly or by word of mouth.

We want a a vertex basis ó a minimal set of vertices with directed paths to all other vertices.

We can build a directedñpath graph for the original graph with the same vertex set and with a directed edge < pi ; pj > added if there is a directed path from pi to pj in the original graph.

Find a Vertex Basis

Find a Vertex Basis