Update graphs.md

Author: djeada

notes/graphs.md

@@ -22,37 +22,37 @@ Viewing these systems as graphs lets us dig deeper into how they work, optimize
 
 Graph theory has its own language, full of terms that make it easier to talk about and define different types of graphs and their features. Here's an explanation of some important terms:
 
-- A **graph** $G$ is a mathematical structure consisting of a set $V(G)$ of vertices (also known as points or nodes) and a set $E(G)$ of pairs of these vertices. These pairs, denoted as $\{x, y\} \in E(G)$, are called edges (or links or lines).
-- If $\{x, y\} \in E(G)$, then $x$ and $y$ are said to be **adjacent**. The number of vertices adjacent to a vertex $v$ defines its degree. Interestingly, the total degrees of all vertices in a graph is always even.
-- A **path** of length $n$ is a series of vertices $v_1 \sim v_2 \sim \dots \sim v_{n+1}$ such that $(v_i, v_{i+1}) \in E(G)$ with all vertices in the sequence being distinct.
-- A **cycle** is a special type of path where $v_1 = v_{n+1}$, and all other vertices from $v_1$ to $v_n$ are unique.
-- Distance represents the length of the shortest path between two vertices. When assessing the distance between vertices $x$ and $y$, you are looking for the most direct route connecting them.
-- A **simple graph** is a graph with no self-loops (edges connected at both ends to the same vertex) and at most one edge between any two vertices.
-- In a **directed graph** (digraph), edges have direction. They are an ordered pair of vertices, often called arcs. In contrast, undirected graphs lack directionality, meaning an edge from vertex $A$ to vertex $B$ is the same as one from $B$ to $A$.
-- A **weighted graph** assigns a weight to each edge, typically a non-negative integer. A binary weight represents the presence (1) or absence (0) of a connection between nodes. A numeric weight indicates the strength or cost of a connection. A normalized weight ensures that all outgoing edges from a node sum to one, often used in probabilistic contexts.
-- In terms of **connectivity**, an undirected graph is termed connected if every vertex pair is linked by some path. In a directed graph, weakly connected means a path exists from vertex $A$ to $B$ or $B$ to $A$ for any vertices $A$ and $B$. Strongly connected means paths exist in both directions between every pair of vertices.
-- **Vertices** $A$ and $B$, if connected by an edge $e$, are neighbors. This edge $e$ is incident to both $A$ and $B$. When two edges share a vertex (like edges from $A$ to $B$ and $B$ to $C$), they are adjacent.
-- The concept of an **isolated vertex** refers to a vertex with no edges connected to it. This means the vertex has a degree of zero since it is not adjacent to any other vertex.
-- **Subgraphs** are smaller graphs formed from a subset of the vertices and edges of a larger graph. If $G$ is a graph, a subgraph $H$ is a graph such that the vertex set of $H$ is a subset of the vertex set of $G$ and the edge set of $H$ is a subset of the edge set of $G$.
-- A **spanning tree** of a graph $G$ is a subgraph that includes all the vertices of $G$ and is a single connected tree. A spanning tree has exactly $|V| - 1$ edges where $|V|$ is the number of vertices in the graph.
-- **Bipartite graphs** are graphs whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. Formally, a graph $G$ is bipartite if there exists a partition $V(G) = V_1 \cup V_2$ where $V_1$ and $V_2$ are disjoint and every edge connects a vertex in $V_1$ to a vertex in $V_2$.
-- A **complete graph** is a simple graph in which there is an edge between every pair of vertices. If a complete graph has $n$ vertices, it is denoted by $K_n$ and has $\frac{n(n-1)}{2}$ edges.
-- **Planar graphs** can be drawn on a plane without any edges crossing. A graph is planar if it can be embedded in the plane, meaning it can be drawn on a flat surface such that its edges intersect only at their endpoints.
-- The **Eulerian path** is a path in a graph that visits every edge exactly once. If such a path exists and starts and ends at different vertices, it is an Eulerian path. If it starts and ends at the same vertex, it is an Eulerian circuit. A graph has an Eulerian circuit if and only if all vertices have an even degree, and it is connected. A graph has an Eulerian path if it has exactly zero or two vertices of odd degree, and it is connected.
-- The **Hamiltonian path** is a path in a graph that visits every vertex exactly once. If such a path exists and starts and ends at different vertices, it is a Hamiltonian path. If it starts and ends at the same vertex, it is a Hamiltonian circuit. Determining whether a Hamiltonian path or circuit exists in a graph is a well-known NP-complete problem.
-- **Graph isomorphism** refers to a condition where two graphs $G$ and $H$ are considered isomorphic if there is a one-to-one correspondence between their vertex sets and their edge sets that preserves adjacency. In other words, $G$ and $H$ are structurally identical but may have different vertex and edge labels.
-- **Degree sequence** is a list of the degrees of the vertices in a graph, usually written in non-increasing order. The degree sequence is a graph invariant, meaning it is the same for any isomorphic graphs.
-- **Graph coloring** is the assignment of labels (colors) to vertices of a graph such that no two adjacent vertices share the same color. The minimum number of colors needed to color a graph $G$ is known as its chromatic number.
-- **Tree** is a connected, acyclic graph. In a tree, any two vertices are connected by exactly one path, and adding any edge to a tree will create a cycle. Trees have several important properties and applications in computer science, particularly in data structures and algorithms.
-- **Forest** is a disjoint set of trees. It is an acyclic graph that may consist of multiple disconnected components, each being a tree.
-- **Cut vertices and cut edges** refer to vertices and edges whose removal increases the number of connected components of the graph. A cut vertex (or articulation point) is a vertex that, if removed, would split the graph into two or more disconnected subgraphs. A cut edge (or bridge) is an edge that, if removed, would increase the number of connected components.
-- **Matching** in a graph is a set of edges without common vertices. A maximum matching is a matching that contains the largest possible number of edges. If every vertex of the graph is incident to exactly one edge of the matching, it is called a perfect matching.
-- **Independent set** is a set of vertices in a graph, no two of which are adjacent. The size of the largest independent set is called the independence number of the graph.
-- **Clique** is a subset of vertices such that every two distinct vertices are adjacent. The size of the largest clique in a graph is called the clique number.
-- **Vertex cover** is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. The size of the smallest vertex cover is called the vertex cover number.
-- **Dominating set** is a set of vertices such that each vertex in the graph is either in the dominating set or adjacent to a vertex in the dominating set. The size of the smallest dominating set is called the domination number.
-- **Planarity testing** determines whether a graph can be drawn in a plane without edge crossings. Kuratowski's theorem provides a criterion for planarity based on the presence of subgraphs homeomorphic to $K_5$ (complete graph on five vertices) or $K_{3,3}$ (complete bipartite graph on six vertices, three in each set).
-- **Graph algorithms** encompass a variety of procedures and techniques used to solve problems related to graphs. Examples include depth-first search (DFS), breadth-first search (BFS), Dijkstra's algorithm for shortest paths, Kruskal's and Prim's algorithms for minimum spanning trees, and the Bellman-Ford algorithm for shortest paths in graphs with negative weights.
+- A *graph* $G$ is a mathematical structure composed of vertices (also known as nodes or points), forming the set $V(G)$, and edges (also called links or lines), forming the set $E(G)$. Each edge connects two distinct vertices, denoted by pairs $\{x, y\} \in E(G)$.
+- When two vertices, say $x$ and $y$, share an edge, they are termed *adjacent*. The count of adjacent vertices to any vertex $v$ defines its *degree*. Notably, summing the degrees of all vertices in a graph always yields an even number.
+- A *path* of length $n$ is a sequence of vertices $v_1 \sim v_2 \sim dots \sim v_{n+1}$, each consecutively connected by edges, with no vertex repeated.
+- A *cycle* is a special path where the first and last vertices are identical, forming a closed loop, and no other vertex is repeated within the cycle.
+- *Distance* in a graph refers to the shortest path length between two vertices. Essentially, it measures how closely two vertices are connected through the minimal number of edges.
+- A *simple graph* is characterized by the absence of self-loops—edges that connect vertices to themselves—and contains no more than one edge between any two distinct vertices.
+- A *directed graph* (or digraph) includes edges with specific directions, known as arcs, represented as ordered pairs of vertices. Conversely, an undirected graph treats connections symmetrically, meaning the connection between vertices $A$ and $B$ is identical to the one from $B$ to $A$.
+- A *weighted graph* assigns numerical values (weights) to edges, typically non-negative integers. Binary weights (0 or 1) indicate connection presence or absence; numeric weights quantify costs or strengths; and normalized weights scale outgoing connections from a vertex to sum to one, commonly used in probability models.
+- Regarding *connectivity*, an undirected graph is *connected* if there's a path linking any pair of vertices. For directed graphs, *weak connectivity* means at least one directional path exists between vertex pairs, while *strong connectivity* demands a path in both directions between every vertex pair.
+- Two vertices connected by an edge are called *neighbors*, and the edge itself is said to be *incident* to these vertices. Edges sharing a common vertex are described as *adjacent* edges.
+- An *isolated vertex* refers to a vertex with no connecting edges, hence having a degree of zero and no neighbors.
+- *Subgraphs* are smaller structures derived from selecting subsets of vertices and edges from a larger graph. Formally, a subgraph $H$ of a graph $G$ has vertex and edge sets entirely contained within those of $G$.
+- A *spanning tree* of a graph $G$ is a subgraph that connects all vertices using the minimal number of edges required, specifically $|V| - 1$ edges, and contains no cycles.
+- *Bipartite graphs* partition vertices into two distinct groups such that no edges connect vertices within the same group. Formally, a graph is bipartite if it can be split into sets $V_1$ and $V_2$ where each edge links vertices across the two sets only.
+- A *complete graph*, denoted by $K_n$, is a simple graph with an edge between every possible pair of vertices. Thus, a complete graph with $n$ vertices contains exactly $\frac{n(n-1)}{2}$ edges.
+- *Planar graphs* are graphs that can be drawn on a flat plane without edge intersections except at vertices. Such graphs can be embedded clearly onto a two-dimensional surface without visual confusion.
+- An *Eulerian path* travels through each edge exactly once. If this path forms a loop, starting and ending at the same vertex, it becomes an *Eulerian circuit*. For a graph to have an Eulerian circuit, all vertices must have even degrees and the graph must be connected; an Eulerian path exists if exactly zero or two vertices have odd degrees.
+- A *Hamiltonian path* traverses every vertex exactly once. If such a path returns to its starting vertex, it becomes a *Hamiltonian circuit*. The decision about the existence of Hamiltonian paths or circuits is notably difficult and classified as an NP-complete problem.
+- *Graph isomorphism* describes two graphs, $G$ and $H$, that are structurally identical despite potentially different vertex or edge labels. Formally, an isomorphism is a one-to-one correspondence preserving adjacency between their vertex sets.
+- The *degree sequence* of a graph lists all vertex degrees, typically ordered from highest to lowest. This sequence serves as a graph invariant, meaning it is consistent across isomorphic graphs.
+- *Graph coloring* involves assigning different labels or "colors" to adjacent vertices. The minimum number of colors required to color a graph without adjacent vertices sharing the same color is called the chromatic number.
+- A *tree* is a connected graph with no cycles. It possesses a unique path between every pair of vertices, and adding any extra edge will inevitably create a cycle. Trees are important in many computational applications, especially algorithms and data structures.
+- A *forest* consists of multiple disconnected trees. It's essentially an acyclic graph that isn't necessarily connected, serving as a generalization of trees.
+- Regarding *connectivity*, a vertex whose removal increases the number of disconnected parts is an *articulation vertex* (or cut vertex). Similarly, an edge whose removal disconnects the graph is called a *bridge* or cut edge.
+- A *matching* in a graph is a set of edges with no shared vertices. A *maximum matching* is the largest possible such set.
+- An *independent set* is a vertex set with no edges connecting any two vertices. The largest size of such a set is known as the independence number.
+- A *clique* is a subset of vertices in which each vertex connects directly to all others. The largest clique size is called the clique number.
+- A *vertex cover* is a collection of vertices such that every graph edge touches at least one vertex from the collection.
+- A *clique* is the opposite concept—vertices in a clique are all mutually adjacent. The largest possible clique size defines the clique number.
+- *Planarity testing* checks if a graph can be drawn without intersecting edges. Kuratowski’s theorem states that non-planarity occurs precisely when a graph contains subgraphs similar to $K_5$ (complete graph of five vertices) or $K_{3,3}$ (complete bipartite graph).
+- Common algorithms on graphs include methods for finding shortest paths (Bellman-Ford, Dijkstra's algorithm), discovering minimum spanning trees (Kruskal’s, Prim’s algorithm), and graph traversal (DFS, BFS). These algorithms help solve important graph-related problems in computing and operations research.
 
 ### Representation of Graphs in Computer Memory