# Prove that a language is decidable

I need some help to prove that the language is decidable.

$$K$$ = {$$N$$ : $$N$$ is a DFA (Sigma = {a, b, c}) and $$L$$($$N$$) contains at least one word in which there is no a}.

It tried to make an algorithm which receives as input a DFA $$N$$ and which determines whether or not this automaton accepts at least one word which contains no a. But it's kinda complex so any help would be appreciated.

A DFA is essentially a directed multi-graph $$G$$ in which a distinguished vertex is marked as the initial state $$s$$, a set $$T$$ of vertices are marked as accepting states, and edges are labelled with the symbols of $$\Sigma$$.
The problem is then equivalent to that of deciding whether there is a path on $$G$$ that goes from $$s$$ to a vertex in $$T$$, and uses only edges that are not labelled with $$a$$.
It suffices to delete all the edges labelled with $$a$$ from $$G$$ and to run any graph visit algorithm from $$s$$. If a vertex in $$T$$ is discovered you can immediately accept. If the visit terminates and no vertex in $$T$$ is discovered, then reject.