Induced subgraph

Gráfico elaborado a partir de un subconjunto de los nodos de otro gráfico y sus aristas

In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges (from the original graph) connecting pairs of vertices in that subset.

Definition

Formally, let ${\displaystyle G=(V,E)}$ be any graph, and let ${\displaystyle S\subseteq V}$ be any subset of vertices of G. Then the induced subgraph ${\displaystyle G[S]}$ is the graph whose vertex set is ${\displaystyle S}$ and whose edge set consists of all of the edges in ${\displaystyle E}$ that have both endpoints in ${\displaystyle S}$.^[1] That is, for any two vertices ${\displaystyle u,v\in S}$, ${\displaystyle u}$ and ${\displaystyle v}$ are adjacent in ${\displaystyle G[S]}$ if and only if they are adjacent in ${\displaystyle G}$. The same definition works for undirected graphs, directed graphs, and even multigraphs.

The induced subgraph ${\displaystyle G[S]}$ may also be called the subgraph induced in ${\displaystyle G}$ by ${\displaystyle S}$, or (if context makes the choice of ${\displaystyle G}$ unambiguous) the induced subgraph of ${\displaystyle S}$.

Examples

Important types of induced subgraphs include the following.

The snake-in-the-box problem concerns the longest induced paths in hypercube graphs

• Induced paths are induced subgraphs that are paths. The shortest path between any two vertices in an unweighted graph is always an induced path, because any additional edges between pairs of vertices that could cause it to be not induced would also cause it to be not shortest. Conversely, in distance-hereditary graphs, every induced path is a shortest path.^[2]
• Induced cycles are induced subgraphs that are cycles. The girth of a graph is defined by the length of its shortest cycle, which is always an induced cycle. According to the strong perfect graph theorem, induced cycles and their complements play a critical role in the characterization of perfect graphs.^[3]
• Cliques and independent sets are induced subgraphs that are respectively complete graphs or edgeless graphs.
• Induced matchings are induced subgraphs that are matchings.
• The neighborhood of a vertex is the induced subgraph of all vertices adjacent to it.

Computation

El problema de isomorfismo de subgrafos inducido es una forma del problema de isomorfismo de subgrafos en el que el objetivo es probar si un gráfico se puede encontrar como un subgrafo inducido de otro. Debido a que incluye el problema de la camarilla como un caso especial, es NP-completo . ^[4]

Referencias

1. Diestel, Reinhard (2006), Teoría de grafos, Textos de posgrado en matemáticas, vol. 173, Springer-Verlag, págs. 3–4, ISBN 9783540261834.
2. Howorka, Edward (1977), "Una caracterización de gráficos hereditarios de distancia", The Quarterly Journal of Mathematics , segunda serie, 28 (112): 417–420, doi :10.1093/qmath/28.4.417, MR  0485544.
3. Chudnovsky, María ; Robertson, Neil ; Seymour, Pablo ; Thomas, Robin (2006), "El teorema del grafo perfecto fuerte", Annals of Mathematics , 164 (1): 51–229, arXiv : math/0212070 , doi :10.4007/annals.2006.164.51, MR  2233847.
4. Johnson, David S. (1985), "La columna de integridad NP: una guía continua", Journal of Algorithms , 6 (3): 434–451, doi :10.1016/0196-6774(85)90012-4, MR  0800733.