Math Problem Statement

Consider the following graph:

\begin{center} \begin{tikzpicture}[scale=1] \tikzstyle{every node}+=[draw,circle,inner sep=0pt,minimum size=20] \node (a) at (0,0) {$a$}; \node (b) at (3,0) {$b$}; \node (c) at (0,3) {$c$}; \node (d) at (3,3) {$d$}; \node (e) at (6,3) {$e$}; \node (f) at (0,6) {$f$}; \node (g) at (3,6) {$g$}; \node (h) at (6,6) {$h$}; \node (i) at (0,9) {$i$}; \node (j) at (3,9) {$j$}; \node (k) at (6,9) {$k$}; \draw (a) edge (b) edge (c); \draw (b) edge (d) edge (e); \draw (c) edge (d) edge (f); \draw (d) edge (e) edge (g); \draw (e) edge (h); \draw (f) edge (g) edge (i); \draw (g) edge (h) edge (j); \draw (h) edge (k); \draw (i) edge (j); \end{tikzpicture} \end{center}

How many connected components are in the subgraph induced by each of the following subsets of the vertices? Please list which vertices are in each connected component.

\begin{enumerate} \item ${a,b,c,d}$ \item ${e,f,g,k}$ \item ${a,b,e,g,h}$ \item ${c,d,f,h}$ \item ${a,b,f,g, k}$ \end{enumerate}

Solution

To find the number of connected components in each subgraph and list the vertices in each connected component, we need to examine each subset and the edges among its vertices based on the given graph structure.

1. Subset: {a,b,c,d}\{a, b, c, d\}

  • Edges: aba \leftrightarrow b, aca \leftrightarrow c, bdb \leftrightarrow d, cdc \leftrightarrow d
  • Connected Components: This subset forms a fully connected subgraph (a single connected component).
  • Connected Component: {a,b,c,d}\{a, b, c, d\}

2. Subset: {e,f,g,k}\{e, f, g, k\}

  • Edges: No edges directly connect all four vertices in this subset.
  • Analysis:
    • ee is isolated as it has no connections with ff, gg, or kk within this subset.
    • ff and gg are connected through an edge.
    • kk is isolated, as it has no edges with ee, ff, or gg.
  • Connected Components:
    • {e}\{e\}
    • {f,g}\{f, g\}
    • {k}\{k\}
  • Total Connected Components: 3

3. Subset: {a,b,e,g,h}\{a, b, e, g, h\}

  • Edges: beb \leftrightarrow e, ghg \leftrightarrow h
  • Analysis:
    • aa is isolated.
    • bb and ee form a connected component.
    • gg and hh form a connected component.
  • Connected Components:
    • {a}\{a\}
    • {b,e}\{b, e\}
    • {g,h}\{g, h\}
  • Total Connected Components: 3

4. Subset: {c,d,f,h}\{c, d, f, h\}

  • Edges: No edges directly connect all four vertices in this subset.
  • Analysis:
    • cc and dd are connected.
    • ff is isolated within this subset.
    • hh is isolated within this subset.
  • Connected Components:
    • {c,d}\{c, d\}
    • {f}\{f\}
    • {h}\{h\}
  • Total Connected Components: 3

5. Subset: {a,b,f,g,k}\{a, b, f, g, k\}

  • Edges: No edges directly connect all five vertices in this subset.
  • Analysis:
    • aa and bb form a connected component.
    • ff and gg form a connected component.
    • kk is isolated.
  • Connected Components:
    • {a,b}\{a, b\}
    • {f,g}\{f, g\}
    • {k}\{k\}
  • Total Connected Components: 3

Summary of Results

  1. Subset {a,b,c,d}\{a, b, c, d\}: 1 connected component: {a,b,c,d}\{a, b, c, d\}
  2. Subset {e,f,g,k}\{e, f, g, k\}: 3 connected components: {e}\{e\}, {f,g}\{f, g\}, {k}\{k\}
  3. Subset {a,b,e,g,h}\{a, b, e, g, h\}: 3 connected components: {a}\{a\}, {b,e}\{b, e\}, {g,h}\{g, h\}
  4. Subset {c,d,f,h}\{c, d, f, h\}: 3 connected components: {c,d}\{c, d\}, {f}\{f\}, {h}\{h\}
  5. Subset {a,b,f,g,k}\{a, b, f, g, k\}: 3 connected components: {a,b}\{a, b\}, {f,g}\{f, g\}, {k}\{k\}

Would you like additional details on any specific subset or related concepts?


Follow-up Questions

  1. How are connected components identified in larger graphs?
  2. What are the applications of connected components in graph theory?
  3. How does the concept of an induced subgraph affect the structure of connected components?
  4. Can we determine the connected components of a subset efficiently?
  5. How would adding more vertices or edges change the number of connected components?

Tip

To quickly find connected components, look for clusters of vertices that share paths, disregarding any isolated nodes or disconnected clusters.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Connected Components
Induced Subgraph

Formulas

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Theorems

Connected Component Theorem

Suitable Grade Level

Grades 11-12