Two definitions of connectedness in graphs

In this short post, we are only dealing with simple graphs G = (V,E) where V is a non-empty set and E \subseteq {\cal P}_2(V) := \{\{a,b\} \subseteq V: a \neq b\}.

When talking about connected graph, one usually has the following definition in mind:

Definition 1. A graph G=(V,E) is said to be connected if for any distinct vertices a,b \in V there is a path connecting a and b. (A  path from a to b is an injective map p: \{0,...n\} \to V for some positive integer n, such that \{p(k), p(k+1)\} \in E for all 0 \leq k < n.)

Note the similarity to path connectedness in topology.

Definition 2. A graph G=(V,E) is said to be connected if for any subset S of V with \emptyset \neq S \neq V there is e \in E such that S \cap e \neq \emptyset \neq (V\setminus S) \cap e.

Again, there is a topological equivalent: we say that a topological space (X,\tau) is connected if no subset S of X with \emptyset \neq S \neq X is both open and closed (“clopen”).

But the analogy of the connectedness in topological spaces and in graphs only goes so far. Recall that every path-connected topological space is connected, but not vice versa. However, and possibly a bit surprisingly even for infinite graphs, we have:

Lemma.  The definitions 1 and 2 above agree.

Proof. It is quite straightforward to show that path-connectedness as in definition 1 implies connectedness as in definition 2. Suppose that G = (V,E) is connected in the sense of definition 2 but not 1. Pick any x \in V and let C = \{x\} \cup \{ v\in V: \textrm{ there is a path from } x \textrm{ to } v\}. (C is the pathwise connected component of x.) It is easy to see that C is a proper subset of V because of our assumption. However, since G is connected in the 2nd sense, there is an edge connecting a point c \in C to some point y \in V\setminus C. But since there is a path from x to c and also \{c, y\} \in E we can extend the path so that it leads from x to y. So y \in C, contradicting y \in V\setminus C.

 

About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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