Graph Theoretic Applications

Graph Theoretic Applications

Dr. G.H.J. Lanel

Introduction to Graphs

Dr. G.H.J. Lanel (USJP) Graph Theoretic Applications Introduction to Graphs 1 / 68

Outline

Outline


Basic Definitions in Graph Theory

Outline1 Basic Definitions in Graph Theory

Handshaking LemmaIsomorphismSome Important Graphs

2 Graph RepresentationAdjacency MatricesAdjacency Lists

3 Paths in Graphs4 Graph Coloring5 Partition Graphs and Their Coloring6 Circuits/Cycles7 Some Properties of Graphs8 Trees9 Unions and Sums of Graphs10 Some Other Properties of Graphs



A graph is a non-empty finite set V of elements called verticestogether with a possibly empty set E of pairs of vertices called edges.



Here are a few examples of graphs:

Vertex set V = {a,b, c,d} and edge set E = {(a,b), (b,d)}Vertex set V = {1,2,3,4} and edge set E = {(2,4)}Vertex set V = {wolf ,goat , cabbage} and edge setE = {(wolf , cabbage)}Vertex set V = {A,B,C} and edge set E = {}















We can draw pictures to represent graphs. Vertices are represented bydots and an edge (v ,w) by an arc that starts from the dot representingv and ends at the dot representing w .

The simplest type of graph is a null graph. It consists of a non-emptyfinite set of vertices and no edges.



We can draw pictures to represent graphs. Vertices are represented bydots and an edge (v ,w) by an arc that starts from the dot representingv and ends at the dot representing w .

The simplest type of graph is a null graph. It consists of a non-emptyfinite set of vertices and no edges.


Basic Definitions in Graph Theory Handshaking Lemma

The degree of a vertex, also called the valence of the vertex, is thenumber of edges incident to it, i.e., the number of edges that have it asan end point.

A simple graph is a graph that has no loops and also does not havemore than one edge between any two vertices.

Handshaking Lemma: The sum of the degrees of the vertices in agraph is twice the number of edges.












Basic Definitions in Graph Theory Isomorphism

Let G1 and G2 be two graphs and let f be a function from the vertex setof G1 to the vertex set of G2.

Suppose that

f is one-to-one and ontof (v) is adjacent to f (w) in G2 if and only if v is adjacent to w in G1



Then we say that the function f is an isomorphism and that the twographs G1 and G2 are isomorphic.

Two graphs G1 and G2 are therefore isomorphic if there is aone-to-one correspondence between vertices of G1 and those of G2with the property that two vertices of G1 are adjacent if and only if theirimages in G2 are adjacent.

If two graphs are isomorphic then as far as we are concerned they arethe same graph.



Then we say that the function f is an isomorphism and that the twographs G1 and G2 are isomorphic.

Two graphs G1 and G2 are therefore isomorphic if there is aone-to-one correspondence between vertices of G1 and those of G2with the property that two vertices of G1 are adjacent if and only if theirimages in G2 are adjacent.

If two graphs are isomorphic then as far as we are concerned they arethe same graph.


Basic Definitions in Graph Theory Some Important Graphs

Directed graphs/Digraphs

The graphs we have been dealing with so far have been symmetric inthe sense that if vertex a is related to vertex b then vertex b is relatedto vertex a.

This will not always be the case as you will discover when we talkabout directed graphs.

Think of a relation between people defined by ” a is related to b if a istaller than b ”. That is surely not going to be symmetric.

In such a case we shall call the edge a directed edge and we shallrefer to a as the initial vertex and b as the terminal vertex.



Directed graphs/Digraphs

The graphs we have been dealing with so far have been symmetric inthe sense that if vertex a is related to vertex b then vertex b is relatedto vertex a.

This will not always be the case as you will discover when we talkabout directed graphs.

Think of a relation between people defined by ” a is related to b if a istaller than b ”. That is surely not going to be symmetric.

In such a case we shall call the edge a directed edge and we shallrefer to a as the initial vertex and b as the terminal vertex.



Complete graphs

A complete graph is a graph where every vertex is adjacent to everyother vertex.

A complete graph on n vertices is denoted by Kn.

So, for example, the complete graph of 5 vertices K5 is given below.

The complete graph K5



Complete graphs

A complete graph is a graph where every vertex is adjacent to everyother vertex.

A complete graph on n vertices is denoted by Kn.

So, for example, the complete graph of 5 vertices K5 is given below.

The complete graph K5



Subgraphs

We say that a graph G1 is a subgraph of a graph G2 if G1 is isomorphicto a graph all of whose vertices and edges are in G2. Note that by ourdefinition a graph is always a subgraph of itself.



Regular Graphs

A graph in which every vertex has the same degree is called aregular graph.If every vertex has degree r then we say the graph is regular ofdegree r .A graph that is regular of degree r is said to be r-regular.All null graphs are regular of degree zero.



Regular Graphs




Regular Graphs




Regular Graphs



Graph Representation






Graph Representation Adjacency Matrices

The adjacency matrix of a graph on n vertices is an nxn matrixA = (aij) in which the entry aij = 1 if there is an edge from vertex i tovertex j and is 0 if there is no edge from vertex i to vertex j .


Graph Representation Adjacency Lists

Another way to represent the edges and vertices of a graph is to usean adjacency list.

The row consists of the label of the vertex followed by a list of thelabels of all the vertices adjacent to it. So, for instance if vertex 2 isadjacent to vertices 1 and 4 the row for vertex 2 would run 2 1 4.


Graph Representation Adjacency Lists

Another way to represent the edges and vertices of a graph is to usean adjacency list.

The row consists of the label of the vertex followed by a list of thelabels of all the vertices adjacent to it. So, for instance if vertex 2 isadjacent to vertices 1 and 4 the row for vertex 2 would run 2 1 4.


Paths in Graphs






Paths in Graphs

We now introduce the idea of a path in a graph.

A path of length n , n ≥ 1 , from vertex v1 to vertex vn is a sequence ofvertices v1, v2, ..., vn such that the initial vertex is v1, and the terminalvertex is vn.


Paths in Graphs

We now introduce the idea of a path in a graph.

A path of length n , n ≥ 1 , from vertex v1 to vertex vn is a sequence ofvertices v1, v2, ..., vn such that the initial vertex is v1, and the terminalvertex is vn.


Paths in Graphs

If a path starts and ends at the same vertex we say the path is aclosed path.

If a path does not visit any vertex more than once it is a simple path.

A path that is both simple and closed is a cycle.


Paths in Graphs

If a path starts and ends at the same vertex we say the path is aclosed path.

If a path does not visit any vertex more than once it is a simple path.

A path that is both simple and closed is a cycle.


Paths in Graphs

Connected Graphs

A vertex v in a graph G is reachable from a vertex u if u = v or there isa path in G from u to v .

A graph G is connected if every vertex in G is reachable from everyother vertex of G. A graph that is not connected is a disconnectedgraph.


Paths in Graphs

Connected Graphs

A vertex v in a graph G is reachable from a vertex u if u = v or there isa path in G from u to v .

A graph G is connected if every vertex in G is reachable from everyother vertex of G. A graph that is not connected is a disconnectedgraph.


Graph Coloring






Graph Coloring

Vertex coloring

Get the graph of a cube. What this does is to color the vertices of thegraph using as few colors as possible and making sure that adjacentvertices always have different colors.

We shall call such a coloring a proper vertex coloring.


Graph Coloring

Vertex coloring

Get the graph of a cube. What this does is to color the vertices of thegraph using as few colors as possible and making sure that adjacentvertices always have different colors.

We shall call such a coloring a proper vertex coloring.


Graph Coloring

The cube, for example, can be properly colored using just two colors.The following figure shows a properly colored cube.

The number of colors used in a proper coloring of a graph is called thechromatic number of the graph. We use the notation χG for thechromatic number of the graph G.


Graph Coloring

The cube, for example, can be properly colored using just two colors.The following figure shows a properly colored cube.

The number of colors used in a proper coloring of a graph is called thechromatic number of the graph. We use the notation χG for thechromatic number of the graph G.


Graph Coloring

Edge coloring

Suppose instead of coloring the vertices of a graph we color the edges.

If we color the edges so that edges with a common end vertex havedifferent colors and we use as few colors as possible we get a properedge coloring.

The number of colors used in a proper edge coloring of a graph iscalled the chromatic index of the graph.We use the notation χ

′

G for the chromatic index of the graph G.


Graph Coloring

Edge coloring




′



Graph Coloring

Edge coloring




′



Partition Graphs and Their Coloring







Bipartite Graphs

A bipartite graph is a graph G whose vertex set V can be partitionedinto two non empty sets V1 and V2 in such a way that every edge of Gjoins a vertex in V1 to a vertex in V2.

An alternative way of thinking about it is that if you were to color thevertices in V1 one color and those in V2 another color then adjacentvertices would have different colors.



Bipartite Graphs

A bipartite graph is a graph G whose vertex set V can be partitionedinto two non empty sets V1 and V2 in such a way that every edge of Gjoins a vertex in V1 to a vertex in V2.

An alternative way of thinking about it is that if you were to color thevertices in V1 one color and those in V2 another color then adjacentvertices would have different colors.



A non-null graph is bipartite if and only if its chromatic number is 2.

A null graph with more than one vertex is trivially bipartite as you canpartition the vertex set any way you like.



Complete Bipartite Graphs

A complete bipartite graph is a graph G whose vertex set V can bepartitioned into two non empty sets V1 and V2 in such a way that everyvertex in V1 is adjacent to every vertex in V2.

If V1 has r vertices and V1 has s vertices then the complete bipartitegraph is written as Kr ,s.



Complete Bipartite Graphs

A complete bipartite graph is a graph G whose vertex set V can bepartitioned into two non empty sets V1 and V2 in such a way that everyvertex in V1 is adjacent to every vertex in V2.

If V1 has r vertices and V1 has s vertices then the complete bipartitegraph is written as Kr ,s.



Star Graphs

Suppose we start with n vertices, choose one special vertex and thendraw edges from the special vertex to every other vertex. The graphwe would obtain is called the star on n vertices, Sn.

Figure shows the star on ten vertices, S10.

The star S10



Star Graphs

Suppose we start with n vertices, choose one special vertex and thendraw edges from the special vertex to every other vertex. The graphwe would obtain is called the star on n vertices, Sn.

Figure shows the star on ten vertices, S10.

The star S10



Tripartite Graphs

In the same way as with the bipartite graphs, if we can divide thevertex set into three disjoint non empty sets V1, V2 and V3 so thatvertices in the same set are not adjacent we get a tripartite graph.

A complete tripartite graph is a tripartite graph where every pair ofvertices that are not in the same set of the partition is adjacent.



Tripartite Graphs

In the same way as with the bipartite graphs, if we can divide thevertex set into three disjoint non empty sets V1, V2 and V3 so thatvertices in the same set are not adjacent we get a tripartite graph.

A complete tripartite graph is a tripartite graph where every pair ofvertices that are not in the same set of the partition is adjacent.



We represent a complete tripartite graph as Kr ,s,t where r is thenumber of vertices in V1, s the number of vertices in V2 and t thenumber of vertices in V3.

Complete tripartite graph K2,3,3


Circuits/Cycles






Circuits/Cycles

A simple circuit on n vertices, n > 2, is a graph with n vertices,x1, x2, ..., xn each of which has degree 2, with xi adjacent to xi+1 fori = 1,2, ...,n − 1, and xn adjacent to x1.

We shall denote a simple circuit (or cycle) on n vertices by Cn.

The circuit graph C7


Circuits/Cycles

A simple circuit on n vertices, n > 2, is a graph with n vertices,x1, x2, ..., xn each of which has degree 2, with xi adjacent to xi+1 fori = 1,2, ...,n − 1, and xn adjacent to x1.

We shall denote a simple circuit (or cycle) on n vertices by Cn.

The circuit graph C7


Circuits/Cycles

Girths

The size of the shortest cycle in a graph is called the girth of the graph.If there is no cycle in the graph the girth is considered to be infinite.


Circuits/Cycles

Wheel Graphs

A wheel on n vertices Wn is a graph with n vertices x1, x2, ..., xn, with x1having degree n − 1 and all the other vertices having degree 3.

The vertex x1 is adjacent to all the other vertices, and fori = 2, ...,n − 1, xi is adjacent to xi+1, and xn−1 is adjacent to x2. Weshall assume that n > 3 in all the cases.


Circuits/Cycles

Wheel Graphs

A wheel on n vertices Wn is a graph with n vertices x1, x2, ..., xn, with x1having degree n − 1 and all the other vertices having degree 3.

The vertex x1 is adjacent to all the other vertices, and fori = 2, ...,n − 1, xi is adjacent to xi+1, and xn−1 is adjacent to x2. Weshall assume that n > 3 in all the cases.


Circuits/Cycles

Euler Circuits

An Euler circuit in a graph G is a circuit that visits each vertex of G anduses every edge of G.

An Euler circuit does not have to be simple, i.e., it can visit the samevertex more than once. It also does not have to be unique. A graphcould have many Euler circuits.

An Euler path on a graph G is a path that visits each vertex of G anduses every edge of G.


Circuits/Cycles

Euler Circuits





Circuits/Cycles

Euler Circuits





Circuits/Cycles

Hamilton Circuits

A Hamilton circuit (also called a Hamiltonian circuit ) on a graph G is acircuit that visits each vertex of G exactly once.

It does not have to use every edge of G. It is named after Sir WilliamHamilton who created a puzzle that challenged one to find a Hamiltoncircuit in a dodecahedron.


Circuits/Cycles

Hamilton Circuits

A Hamilton circuit (also called a Hamiltonian circuit ) on a graph G is acircuit that visits each vertex of G exactly once.

It does not have to use every edge of G. It is named after Sir WilliamHamilton who created a puzzle that challenged one to find a Hamiltoncircuit in a dodecahedron.


Circuits/Cycles

N-dimensional Hypercubes

We define a one dimensional hypercube (the 1-cube) as a graph withtwo vertices and one edge. Recursively, if you have the (n − 1)-cube,you get the n-cube by taking two isomorphic copies of the (n− 1)-cubeand adding edges between corresponding vertices.

Thus the 2-cube is a rectangle, the 3-cube is the usual 3-dimensionalcube, the 4-cube is as shown in figure 1 and the 5-cube is as shown infigure 2 below.

Fig: 1 Fig: 2Dr. G.H.J. Lanel (USJP) Graph Theoretic Applications Introduction to Graphs 38 / 68

Circuits/Cycles

N-dimensional Hypercubes

We define a one dimensional hypercube (the 1-cube) as a graph withtwo vertices and one edge. Recursively, if you have the (n − 1)-cube,you get the n-cube by taking two isomorphic copies of the (n− 1)-cubeand adding edges between corresponding vertices.

Thus the 2-cube is a rectangle, the 3-cube is the usual 3-dimensionalcube, the 4-cube is as shown in figure 1 and the 5-cube is as shown infigure 2 below.

Fig: 1 Fig: 2Dr. G.H.J. Lanel (USJP) Graph Theoretic Applications Introduction to Graphs 38 / 68

Some Properties of Graphs







Independent Sets of Vertices

A set I of vertices in a simple graph G = (V ,E) is an independent setin G if no two vertices in I are adjacent.

If an independent set is not contained in any other independentset then we say it is a maximal independent set.

An independent set whose size is greater than or equal to that ofevery other independent set is called a maximum independentset. In many cases a graph has more than one maximumindependent set.



Independent Sets of Vertices

A set I of vertices in a simple graph G = (V ,E) is an independent setin G if no two vertices in I are adjacent.

If an independent set is not contained in any other independentset then we say it is a maximal independent set.

An independent set whose size is greater than or equal to that ofevery other independent set is called a maximum independentset. In many cases a graph has more than one maximumindependent set.



In above figures the red vertices form maximum independent setsfor the Petersen graph and the cube.

For a graph G, the size of a maximum independent set in G iscalled the independence number of G, written α(G).



In above figures the red vertices form maximum independent setsfor the Petersen graph and the cube.

For a graph G, the size of a maximum independent set in G iscalled the independence number of G, written α(G).



Cliques

A clique of a graph G is a complete subgraph of G.

If a clique is not contained in any other clique in G then we say it is amaximal clique.

A clique whose size is greater than or equal to that of every otherclique in G is called a maximum clique.

In many cases a graph has more than one maximum clique.



Cliques

A clique of a graph G is a complete subgraph of G.

If a clique is not contained in any other clique in G then we say it is amaximal clique.

A clique whose size is greater than or equal to that of every otherclique in G is called a maximum clique.

In many cases a graph has more than one maximum clique.



In the above figure the red vertices form a maximum clique but it iseasy to see that there are other maximum cliques in the graph.

Also, note that the central vertex belongs to multiple maximum cliques.

For a graph G, the size of a maximum clique in G is called the cliquenumber of G, written ω(G).













Split Graphs

Let G be a graph with vertex set V . If we can express the set V as aunion of two disjoint sets of vertices C and I where C is a clique and Iis an independent set then we say that G is a split graph.

Quite often there is more than one way to choose the independent setand the clique so the split is not necessarily unique.

Suppose a graph can be split into a clique C and an independent set Ias above and suppose further that each vertex in the independent setis adjacent to every vertex of the clique. Then we say that G is acomplete split graph.



Split Graphs






Split Graphs






We shall write a complete split graph as Splitn,c , where n is thenumber of vertices in the graph and c is the number of vertices in theclique.

Split7,1 Split13,6


Trees






Trees

A tree is a connected graph that has no circuits as subgraphs.

A graph that has no circuits as subgraphs but is not necessarilyconnected is a forest .


Trees

Tree Traversal

Quite often when data is held in a tree structure, we want to traversethe tree. This means we want to, one by one, examine all of thevertices of the tree. We also want to be sure that we did not omit anyof the vertices. One strategy is the breadth first traversal .


Unions and Sums of Graphs







There are several ways to combine two graphs to get a third one.

Suppose we have graphs G1 and G2 and suppose that G1 has vertexset V1 and edge set E1, and that G2 has vertex set V2 and edge set E2.

The union of the two graphs, written G1 ∪G2 will have vertex setV1 ∪ V2 and edge set E1 ∪ E2.The sum of two graphs G1 and G2, written G1 + G2, is obtained byfirst forming the union G1 ∪G2 and then making every vertex of G1adjacent to every vertex of G2.


















For instance, choose the null graph N1 and the complete graph K5.

Union N1 ∪ K5 Sum N1 + K5


Some Other Properties of Graphs







Vertex Connectivity

A vertex v in a connected graph is an articulation point if the removalof v and all edges with v as an end-point from the graph would leave adisconnected graph.

In some literature a vertex is considered to be an articulation point if itsremoval would increase the number of connected components of thegraph. Suppose you have a connected graph. A set of vertices whoseremoval would disconnect the graph is called a vertex cut .

The vertex connectivity of a graph is the size of its smallest vertexcut(s).

If the vertex connectivity of a graph of a graph is k , then we say thegraph is k-vertex connected .

The complete graphs do not have vertex cuts. By convention acomplete graph on n vertices is considered to be (n − 1)-vertexconnected.



Vertex Connectivity








Vertex Connectivity








Vertex Connectivity








Vertex Connectivity








Edge Connectivity

An edge in a connected graph is a bridge edge if the removal of thatedge from the graph would leave a disconnected graph.

Suppose you have a connected graph. A set of edges whose removalwould disconnect the graph is called an edge cut .

The edge connectivity of a graph is the size of its smallest edge cut(s).

If the edge connectivity of a graph of a graph is k , then we say thegraph is k-edge connected .



Edge Connectivity







Edge Connectivity







Edge Connectivity







Complements of Graphs

The complement ∼G of a graph G is a graph with the same vertex setas G and with the property that two vertices are adjacent in ∼G if andonly if they are not adjacent in G.



Ramsey Numbers

Consider the party problem which asks how many guests you need toinvite to a party to guarantee that either at least three of your guestsknow each other or at least three do not know each other.

Let us represent the guests by vertices of a graph and have twovertices adjacent if the guests those vertices represent know eachother.

Having three or more guests who know each other would mean wehave a clique of size three or greater while having three or moreguests who don’t know each other would mean we have anindependent set of size three or greater.



Ramsey Numbers






Ramsey Numbers






We can restate the problem as follows:

How many vertices does a graph G need to have to guarantee that iteither has clique number ω(G)≥ 3 or independence number α(G)≥ 3.

If m and n are positive integers, the Ramsey number R(m,n) is thesmallest number r such that if a graph G has r or more vertices theneither ω(G)≥ m or α(G)≥ n. Our party problem boils down to findingR(3,3).













Prisms

In general if you have a graph G, we shall define the prism operatorPrism(G) to give the graph obtained by taking two isomorphic copiesof G and placing edges between vertices that correspond under theisomorphism. So, for example, the n-cube is Prism( (n − 1)− cube ).

Figure : The 4-cube is Prism(3− cube)



Laces

A lace Lcn,r , also known as a circulant graph, is a graph on n verticesx0, x1, ..., xn−1, with vertex xi adjacent to vertex xj , if j = (i + r) mod n.

Figure : The lace Lc12,2



Line Graphs

The line graph L(G) of a graph G is a graph that has the followingproperties:

1 There is a vertex in L(G) for every edge of G2 Two vertices of L(G) are adjacent if and only if they correspond to

two edges of G with a common end vertex.



Line Graphs






Line Graphs






Grids



Spanning Trees

A spanning tree for a connected graph G is a subgraph of G that is atree and whose vertex set is all the vertices of G.

If G is a tree then it has only one spanning tree, G itself. If G is not atree it has more than one spanning tree.



Planar graphs

When drawing connected graphs one is naturally lead to the questionof crossing edges. One says that a graph is planar if it can be drawn(or represented) without crossing.

The above graphs represent K3,3 (not planar) and K4 (planar).



Dual Graphs

Consider the graph grid3,4. It looks like previously given grid, exceptwe have colored the seven regions defined by the graph (the outsidecounts as a region too).

The seven regions of grid3,4

The dual of a plane graph G is obtained as follows: for each region ofthe graph G we have a vertex in dual(G). If two regions of the graph Ghave a common edge as a border then the corresponding vertices indual(G) will be adjacent.



Dual Graphs

Consider the graph grid3,4. It looks like previously given grid, exceptwe have colored the seven regions defined by the graph (the outsidecounts as a region too).

The seven regions of grid3,4

The dual of a plane graph G is obtained as follows: for each region ofthe graph G we have a vertex in dual(G). If two regions of the graph Ghave a common edge as a border then the corresponding vertices indual(G) will be adjacent.



Weighted Graphs, Shortest Paths

Recall that in the adjacency matrix of a graph we have a 1 in positioni , j if vertex i is adjacent to vertex j . This allows us to express whethertwo vertices are related or not. However, we sometimes want to saymore.

Suppose, for example the vertices represent cities and we have anedge between two vertices if we can get an airline flight from one cityto the other. We might want to include more information like how longthe flight takes. This brings us to the idea of a weighted graph. This isa graph in which each edge is assigned a number, called a weight.



Weighted Graphs, Shortest Paths

Recall that in the adjacency matrix of a graph we have a 1 in positioni , j if vertex i is adjacent to vertex j . This allows us to express whethertwo vertices are related or not. However, we sometimes want to saymore.

Suppose, for example the vertices represent cities and we have anedge between two vertices if we can get an airline flight from one cityto the other. We might want to include more information like how longthe flight takes. This brings us to the idea of a weighted graph. This isa graph in which each edge is assigned a number, called a weight.



Another way to add information to a graph is by labeling the vertices sothat the labels include that information rather than just 1,2,3, ... ora,b, c, .... We would call such a graph a labeled graph.

Once we have a weighted graph we can ask a lot of questions. Forexample we might want to know which path from a vertex v to anothervertex w has the least weight. The weight of a path will be the sum ofthe weights of the edges in the path.

Since we often think of the weights as representing distance we callthis the shortest path.



Another way to add information to a graph is by labeling the vertices sothat the labels include that information rather than just 1,2,3, ... ora,b, c, .... We would call such a graph a labeled graph.

Once we have a weighted graph we can ask a lot of questions. Forexample we might want to know which path from a vertex v to anothervertex w has the least weight. The weight of a path will be the sum ofthe weights of the edges in the path.

Since we often think of the weights as representing distance we callthis the shortest path.



Minimal Spanning Trees

We already know that a graph could have several spanning trees.Suppose we look at all the spanning trees of a weighted graph andrank them by the total weight of their edges.

The tree with least weight, called the Minimal Spanning Tree, isparticularly important. Actually, since we could have ties it is possiblefor a graph to have more than one minimal spanning tree.

Imagine the weighted graph to represent a railroad system with theweights being the cost of maintaining the tracks. Now suppose youwanted to minimize the maintenance costs without disconnecting thegraph.

Why would the minimal spanning tree be of interest? If the graph isdisconnected you can’t, of course, have a minimal spanning tree(why?) but the union of the minimal spanning trees of the individualcomponents is called a Minimal Spanning Forest .
























End!


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