an edge of a graph that joins a vertex to itself is called a loop.
an edge of a graph that joins a vertex to itself is called a loop.
an edge of a graph that joins a vertex to itself is called a?
a loop
once a dummy initial vertex with an edge to every initial vertex and a dummy accepting vertex with an edge from every accepting vertex are added, checking if there is an accepting computation only requires to check if there is a path from the initial vertex to the accepting vertex, which is the reachability problem.









































a computation is said to be unambiguous if there exists at most one path from an initial vertex to an accepting vertex.









































a cycle in the graph corresponds to an infinite loop in the computation.









































section::::size of the graph.
configuration graph
in a line graph "l"("g"), each vertex of degree "k" in the original graph "g" creates "k(k-1)/2" edges in the line graph. for many types of analysis this means high-degree nodes in "g" are over-represented in the line graph "l"("g"). for instance, consider a random walk on the vertices of the original graph "g". this will pass along some edge "e" with some frequency "f". on the other hand, this edge "e" is mapped to a unique vertex, say "v", in the line graph "l"("g"). if we now perform the same type of random walk on the vertices of the line graph, the frequency with which "v" is visited can be completely different from "f". if our edge "e" in "g" was connected to nodes of degree "o(k)", it will be traversed "o(k)" more frequently in the line graph "l"("g"). put another way, the whitney graph isomorphism theorem guarantees that the line graph almost always encodes the topology of the original graph "g" faithfully but it does not guarantee that dynamics on these two graphs have a simple relationship. one solution is to construct a weighted line graph, that is, a line graph with weighted edges. there are several natural ways to do this. for instance if edges "d" and "e" in the graph "g" are incident at a vertex "v" with degree "k", then in the line graph "l"("g") the edge connecting the two vertices "d" and "e" can be given weight "1/(k-1)". in this way every edge in "g" (provided neither end is connected to a vertex of degree '1') will have strength "2" in the line graph "l"("g") corresponding to the two ends that the edge has in "g". it is straightforward to extend this definition of a weighted line graph to cases where the original graph "g" was directed or even weighted. the principle in all cases is to ensure the line graph "l"("g") reflects the dynamics as well as the topology of the original graph "g".
line graph