Given a graph with 6 vertices of degrees 0 2 2 4 4 4, in what ways may it be drawn? Simple and connected or some combination of?
Obviously it can't be connected due to the vertex with degree 0, but what can I do with the remaining 5 vertices? The graph would have 8 edges since $\frac{\sum deg(v_i)}{2} = 8$.
I noticed that $K_5$ has 10 edges which means that each vertex has deg=4. You take away any edge and you end up with 2 edges having deg=3. At this point the only way to get desired set of degrees is to remove an edge between the 2 vertices having deg=3. But we already removed the only edge between them, so the graph of the remaining 5 vertices is either not simple or not connected.
So at this point I drew a graph where each vertex has a loop and 3 vertices are connected in a triangle, giving the desired set of degrees and number of edges.
Is this the only solution and is there a better way to approach the problem other than trial and error in drawing the graph?