Component

A component, or rather, connected component, of a graph is a connected subgraph of that graph whose vertices are isolated from the rest of the graph. Intuitively, it is a portion of the graph that cannot be walked out of.

Components always include all the edges in the original graph that its vertices participate in. Really, it is the set of vertices that describe each component, because all the edges they participate in are assumed to be included.

An isolated vertex is a vertex with degree \(0.\) That means there are no edges incident to it. An isolated vertex counts as a component all by itself.

No \(2\) components of a graph share vertices in common. Their sets of vertices are disjoint.

⚠ If you add even one extra vertex to a component, it becomes a disconnected graph.

When \(2\) vertices are part of the same component of a graph, they are said to be connected because a walk must exist between them.

Logic & Proofs

Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form

Set Theory

Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof

Functions

Function • Floor function • Ceiling function • Inverse function

Algorithms

Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm

Relations

Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class

Number Theory

Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm

Induction

Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition

Combinatorics

Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle

Graph Theory

Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm

Recursion

Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort