12.3: Paths and Cycles
- Page ID
- 60135
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recall the definition of a walk. As we saw in Example 12.2.1, the vertices and edges in a walk do not need to be distinct.
There are many circumstances under which we might not want to allow edges or vertices to be re-visited. Efficiency is one possible reason for this. We have a special name for a walk that does not allow vertices to be re-visited.
Definition: Path
A walk in which no vertex appears more than once is called a path.
Notation
For \(n ≥ 0\), a graph on \(n + 1\) vertices whose only edges are those used in a path of length \(n\) (which is a walk of length \(n\) that is also a path) is denoted by \(P_n\). (Notice that \(P_0 \cong K_1\) and \(P_1 \cong K_2\).)
Notice that if an edge were to appear more than once in a walk, then both of its endvertices would also have to appear more than once, so a path does not allow vertices or edges to be re-visited.
Example \(\PageIndex{1}\)
In the graph
\((a, f, c, h)\) is a path of length \(3\). However, \((a, f, c, h, d, f)\) is not a path, even though no edges are repeated, since the vertex \(f\) appears twice. Both are walks.
Proposition \(\PageIndex{1}\)
Suppose that \(u\) and \(v\) are in the same connected component of a graph. Then any \(u − v\) walk of minimum length is a path. In particular, if there is a \(u − v\) walk, then there is a \(u − v\) path.
- Proof
-
Since \(u\) and \(v\) are in the same connected component of a graph, there is a \(u − v\) walk.
Towards a contradiction, suppose that we have a \(u − v\) walk of minimum length that is not a path. By the definition of a path, this means that some vertex \(x\) appears more than once in the walk, so the walk looks like:
\[(u = u_1, . . . , u_i = x, . . . , u_j = x, . . . , u_k = v),\]
and \(j > i\). Observe that the following is also a \(u − v\) walk:
\[(u = u_1, . . . , u_i = x, u_{j+1}, u_{j+2}, . . . , u_k = v).\]
Since consecutive vertices were adjacent in the first sequence, they are also adjacent in the second sequence, so the second sequence is a walk. The length of the first walk is \(k − 1\), and the length of the second walk is \(k − 1 − (j − i)\). Since \(j > i\), the second walk is strictly shorter than the first walk. In particular, the first walk was not a \(u − v\) walk of minimum length. This contradiction serves to prove that every \(u − v\) walk of minimum length is a path.
This allows us to prove another interesting fact that will be useful later.
Proposition \(\PageIndex{2}\)
Deleting an edge from a connected graph can never result in a graph that has more than two connected components.
- Proof
-
Let \(G\) be a connected graph, and let \(u_v\) be an arbitrary edge of \(G\). If \(G \setminus \{u_v\}\) is connected, then it has only one connected component, so it satisfies our desired conclusion. Thus, we assume in the remainder of the proof that \(G \setminus \{u_v\}\) is not connected.
Let \(G_u\) denote the connected component of \(G \setminus \{u_v\}\) that contains the vertex \(u\), and let \(G_v\) denote the connected component of \(G \setminus \{u_v\}\) that contains the vertex \(v\). We aim to show that \(G_u\) and \(G_v\) are the only connected components of \(G \setminus \{u_v\}\).
Let \(x\) be an arbitrary vertex of \(G\), and suppose that \(x\) is a vertex that is not in \(G_u\). Since \(G\) is connected, there is a \(u − x\) walk in \(G\), and therefore by Proposition 12.3.1 there is a \(u − x\) path in \(G\). Since \(x\) is not in \(G_u\), this \(u−x\) path must use the edge \(u−v\), so must start with this edge since \(u\) only occurs at the start of the path. Therefore, by removing the vertex \(u\) from the start of this path, we obtain a \(v − x\) path that does not use the vertex \(u\). This path cannot use the edge \(u_v\), so must still be a path in \(G \setminus \{u_v\}\). Therefore \(x\) is a vertex in \(G_v\).
Since \(x\) was arbitrary, this shows that every vertex of \(G\) must be in one or the other of the connected components \(G_u\) and \(G_v\), so there are at most two connected components of \(G \setminus \{u_v\}\). Since \(u_v\) was an arbitrary edge of \(G\) and \(G\) was an arbitrary connected graph, this shows that deleting any edge of a connected graph can never result in a graph with more than two connected components.
A cycle is like a path, except that it starts and ends at the same vertex. The structures that we will call cycles in this course, are sometimes referred to as circuits.
Definition: Cycle
A walk of length at least \(1\) in which no vertex appears more than once, except that the first vertex is the same as the last, is called a cycle.
Notation
For \(n ≥ 3\), a graph on \(n\) vertices whose only edges are those used in a cycle of length \(n\) (which is a walk of length \(n\) that is also a cycle) is denoted by \(C_n\).
The requirement that the walk have length at least \(1\) only serves to make it clear that a walk of just one vertex is not considered a cycle. In fact, a cycle in a simple graph must have length at least \(3\).
Example \(\PageIndex{2}\)
In the graph from Example 12.3.1, \((a, e, f, a)\) is a cycle of length \(3\), and \((b, g, d, h, c, f, b)\) is a cycle of length \(6\).
Here are drawings of some small paths and cycles:
We end this section with a proposition whose proof will be left as an exercise.
Proposition \(\PageIndex{3}\)
Suppose that \(G\) is a connected graph. If \(G\) has a cycle in which \(u\) and \(v\) appear as consecutive vertices (so \(u_v\) is an edge of \(G\)) then \(G \setminus \{u_v\}\) is connected.
Exercise \(\PageIndex{1}\)
1) In the graph
(a) Find a path of length \(3\).
(b) Find a cycle of length \(3\).
(c) Find a walk of length \(3\) that is neither a path nor a cycle. Explain why your answer is correct.
2) Prove that in a graph, any walk that starts and ends with the same vertex and has the smallest possible non-zero length, must be a cycle.
3) Prove Proposition 12.3.3.
4) Prove by induction that if every vertex of a connected graph on \(n ≥ 2\) vertices has valency \(1\) or \(2\), then the graph is isomorphic to \(P_n\) or \(C_n\).
5) Let \(G\) be a (simple) graph on \(n\) vertices. Suppose that \(G\) has the following property: whenever \(u \nsim v\), \(dG(u) + dG(v) ≥ n − 1\). Prove that \(G\) is connected.