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‎Chapters/Chapter2/chapter2.md‎

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## Basic definitions
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There is a great set of mathematical problems that can be solved using graph models. Graphs are vastly used in all kind of computer science problems.
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Graphs are discrete mathematical structures that consist of a set of vertices \(also called nodes\) and a set of edges that connect those vertices.
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In computer science is more commonly used the term _node_ instead of _vertex_. In this booklet the term node is going to be the one used.
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There are multiple types of graphs according if the edges are directed or undirected, if the edges are weighted or unweighted and so on.
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In this chapter, basic graph concepts and graphs types are going to be explained to ease the following of the algorithms.
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### Type of Graphs
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#### Directed Graph
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A directed graph is a type of graph in which every edges has a direction: an ingoing node and an outgoing node. The most commonly way of representing a graph is to draw it.
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A directed graph can de drawn like in Figure *@directed@*:
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![A directed graph.](figures/directed_graph.pdf width=30&label=directed)
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#### Undirected Graph
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An undirected graph is a type of graph in which the edges does not have a direction. They can be drawn without a line that does not have any arrow heads. It is
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understood that the graph has no direction if the direction is not specified explicitly as shown in Figure *@undirected@*.
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![An undirected graph.](figures/undirected_graph.pdf width=30&label=undirected)
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#### Weighted Graph
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A weighted graph is a graph that each of its edges has an associated weight \(see Figure *@weighted@*\). In real life examples, the weights can represent several things.
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For example, a graph can be a map in which the nodes represent cities and the edges represent the distance between those cities.
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![A weighted graph: edges have a weight.](figures/weighted_graph.pdf width=40&label=weighted)
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#### Connected Graph
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A connected graph is an undirected graph in which exists a path for every pair of nodes.
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For example, Figure *@connected@* represents a connected graph because from any node you can get to any node.
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![A connected graph: all the nodes are reachable from any others.](figures/connected_graph.pdf width=40&label=connected)
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But, Figure *@directed2@* is a disconnected graph because the nodes F and G are isolated from the rest.
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![A disconnected graph: some nodes are not reachable from others.](figures/disconnected_graph.pdf width=48&label=directed2)
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#### Bipartite Graph
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A bipartite graph is a graph whose set of vertices can be split into two subsets A and B in such a way that each edge of the graph joins a vertex in A and a vertex in B.
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A complete bipartite graph is a bipartite graph in which each vertex in A is joined to each vertex in B by just one edge.
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### Graph Cycle
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A graph cycle is a sequence of adjacent nodes in which all nodes are different except of the first and the last one.
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That means, a graph cycle is a path that ends and starts in the same node without repeating any other node and it has a size grater than 3.
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For example, in Figure *@cycle1@* there is a cycle between nodes _A, B, D_.
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![A Cycle in a graph.](figures/cycle_in_a_graph.pdf width=40&label=cycle1)
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But, in Figure *@cycle2@* there is no cycle between from node A to C because the node D has to be traveled twice.
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Nevertheless, there is a cycle between node D, B and C.
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![In a directed graph, direction is impacting cycle presence. ](figures/not_a_cycle.pdf width=52&label=cycle2)
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#### Directed Acyclic Graph \(DAG\)
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Like the name suggests, a directed acyclic graph is a directed graph that does not have any cycles \(as shown in Figure *@DAG@*\).
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![A directed Acyclic Graph.](figures/dag.pdf width=35&label=DAG)
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#### Strongly Connected Graph
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Unlike the Connected Graph, a Strongly Connect is a **directed** graph in which there is a path for every pair of nodes. Figure *@strongly1@* is a strongly connected graph.
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![A strongly connected graph.](figures/strongly_connected_graph.pdf width=30&label=strongly1)
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Figure *@strongly2@* is not strongly connected because it is not possible to reach node D from node A. However, if the directions of the graph are deleted, the graph becomes a
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**undirected** connected graph. For that reason, Figure *@strongly2@* is called a weakly connected graph.
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![A weakly connected graph.](figures/not_strongly_connected_graph.pdf width=30&label=strongly2)
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#### Strongly Connected Component
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A strongly connected component of a directed graph is the maximal subgraph that is strongly connected. In the figure there are three strongly connected components in the graph:
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$\{A, B, C\}$, $\{F, E\}$, $\{D\}$.
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![Graph with three strongly connected components](figures/strongly_connected_components.pdf width=50)
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### Tree
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A tree is a connected graph without cycles. That means that there is only one path between every pair of vertices. But, in the computer science context, normally a tree is
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represented as a **directed** graph. In that case, the definition will be that a **directed** tree is a directed acyclic graph in which every node has only one
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incoming \(parent\) node \(See Figure *@directedTree@*\). If you remove the direction of the directed acyclic graph the reaming graph has to be an **undirected** tree \(See Figure *@undiTree@*\).
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![A tree: a connected graph without cycles.](figures/tree.pdf width=40&label=undiTree)
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![A directed tree.](figures/directed_tree.pdf width=40&label=directedTree)
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### Conclusion
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These definitions set the stage for the algorithms that we will now describe.
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Identifying clearly the kind of graphs an algorithm is applied on is key because the working hypotheses
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are really important.
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## Basic Definitions
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There is a great set of mathematical problems that can be solved using graph models. Graphs are vastly used in all kind of computer science problems.
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Graphs are discrete mathematical structures that consist of a set of vertices \(also called nodes\) and a set of edges that connect those vertices.
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In computer science is more commonly used the term _node_ instead of _vertex_. In this booklet, the term node is going to be the one used.
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There are multiple types of graphs according if the edges are directed or undirected, if the edges are weighted or unweighted and so on.
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In this chapter, basic graph concepts and graphs types are going to be explained to ease the following of the algorithms.
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### Type of Graphs
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#### Directed Graph
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A directed graph (a *digraph*, for short) is a type of graph in which every edges has a direction: an ingoing node and an outgoing node. The most commonly way of representing a graph is to draw it. A directed graph can be drawn like in Figure *@directed@*.
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![A directed graph.](figures/directed_graph.pdf width=30&label=directed)
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#### Undirected Graph
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An undirected graph is a type of graph in which the edges does not have a direction. They can be drawn without a line that does not have any arrow heads. It is understood that the graph has no direction if the direction is not specified explicitly as shown in Figure *@undirected@*.
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![An undirected graph.](figures/undirected_graph.pdf width=30&label=undirected)
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#### Weighted Graph
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A weighted graph is a graph that each of its edges has an associated weight \(see Figure *@weighted@*\). In real life examples, the weights can represent several things.
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For example, a graph can be a map in which the nodes represent cities and the edges represent the distance between those cities.
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![A weighted graph: edges have a weight.](figures/weighted_graph.pdf width=40&label=weighted)
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#### Connected Graph
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A connected graph is an undirected graph in which exists a path for every pair of nodes.
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For example, Figure *@connected@* represents a connected graph because from any node you can get to any node.
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![A connected graph: all the nodes are reachable from any others.](figures/connected_graph.pdf width=40&label=connected)
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But, Figure *@directed2@* is a disconnected graph because the nodes F and G are isolated from the rest.
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![A disconnected graph: some nodes are not reachable from others.](figures/disconnected_graph.pdf width=48&label=directed2)
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#### Bipartite Graph
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A bipartite graph is a graph whose set of nodes can be split into two subsets $A$ and $B$ in such a way that each edge of the graph joins a node in $A$ and a node in $B$. In Figure @bipartite@, the two subsets are $\{A1, A2, A3, A4, A5\}$ and $\{B1, B2, B3, B4\}$.
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A *complete* bipartite graph is a bipartite graph in which each node in A is joined to each node in B by just one edge.
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![A bipartite graph: the nodes are split into two subsets.](figures/bipartite.png width=48&label=bipartite)
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### Graph Cycle
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A graph cycle is a sequence of adjacent nodes in which all nodes are different except of the first and the last one. That means, a graph cycle is a path that ends and starts in the same node without repeating any other node and it has a size greater than 3.
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For example, in Figure *@cycle1@* there is a cycle between nodes _A, B, D_.
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![A cycle in a graph.](figures/cycle_in_a_graph.pdf width=40&label=cycle1)
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But in Figure *@cycle2@*, there is no cycle from node A to C because the node D has to be traveled twice. Nevertheless, there is a cycle between nodes D, B and C.
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![In a directed graph, direction is impacting cycle presence.](figures/not_a_cycle.pdf width=52&label=cycle2)
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#### Directed Acyclic Graph \(DAG\)
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Like the name suggests, a directed acyclic graph is a directed graph that does not have any cycles \(as shown in Figure *@DAG@*\).
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![A Directed Acyclic Graph (DAG).](figures/dag.pdf width=35&label=DAG)
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#### Strongly Connected Graph
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Unlike the connected graph, a strongly connected graph is a **directed** graph in which there is a path for every pair of nodes. Figure *@strongly1@* is a strongly connected graph.
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![A strongly connected graph.](figures/strongly_connected_graph.pdf width=30&label=strongly1)
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Figure *@strongly2@* is not strongly connected because it is not possible to reach node D from node A. However, if the directions of the graph are deleted, the graph becomes an **undirected** connected graph. For that reason, Figure *@strongly2@* is called a weakly connected graph.
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![A weakly connected graph.](figures/not_strongly_connected_graph.pdf width=30&label=strongly2)
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#### Strongly Connected Component
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A strongly connected component of a directed graph is the maximal subgraph that is strongly connected. In the figure there are three strongly connected components in the graph:
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$\{A, B, C\}$, $\{F, E\}$, $\{D\}$.
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![Graph with three strongly connected components](figures/strongly_connected_components.pdf width=50)
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### Tree
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A tree is a connected graph without cycles. That means that there is only one path between every pair of vertices. But, in the computer science context, normally a tree is
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represented as a **directed** graph. In that case, the definition will be that a **directed** tree is a directed acyclic graph in which every node has only one incoming \(parent\) node \(See Figure *@directedTree@*\). If you remove the direction of the directed acyclic graph, the remaining graph has to be an **undirected** tree \(See Figure *@undiTree@*\).
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![A tree: a connected graph without cycles.](figures/tree.pdf width=40&label=undiTree)
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![A directed tree.](figures/directed_tree.pdf width=40&label=directedTree)
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### Conclusion
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These definitions set the stage for the algorithms that we will now describe.
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Identifying clearly the kind of graphs an algorithm is applied on is key because the working hypotheses
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are really important.

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