User blog:Hyp cos/Are these functions well-defined?

Here are two extensions of subcubic graph function, but I don't know whether they are well-defined.

Minor relationships
Graph A is called a graph minor of graph B if A can be obtained from B by these 3 operations: Graph A is called a topological minor of graph B if A can be obtained from B by these 3 operations: For (unlabeled) subcubic graphs, graph minor and topological minor are the same. Graph minor needs operations merging two vertices into one, so it's "unfriendly" to labeled vertices. Topological minor needs operations merging two vertices into one, so it's "unfriendly" to labeled edges.
 * 1) Delete an edge;
 * 2) Delete an isolated vertex;
 * 3) Edge contraction - for an edge ab, delete this edge, and replace vertex a and b by one vertex connecting to the edges which a or b formerly connect to. (If a and b are linked by another edge, it will become a self-loop after edge contraction.)
 * 1) Delete an edge;
 * 2) Delete an isolated vertex;
 * 3) Smoothing - for a (degree=2) vertex a which links to b and c, delete vertex a, and replace edge ab and ac by one edge bc.

ESCG
ESCG (shorthand for edge-labeled subcubic graph) function is defined as follows. ESCG(n) is the maximal length of sequences of graphs (whose edges are colored using n labels (or n-colored)) {G1, G2, ..., GN} such that Then, is ESCG function well-defined? In another word, are this kind of sequences always finite for all n?
 * For all i, Gi has at most i vertices.
 * For all i, the degree of every vertex of Gi is at most 3.
 * For all i < j, Gi is not graph minor of Gj.

VSCG
VSCG (shorthand for vertex-labeled subcubic graph) function is defined as follows. VSCG(n) is the maximal length of sequences of graphs (whose vertices are colored using n labels (or n-colored)) {G1, G2, ..., GN} such that Then, is VSCG function well-defined? In another word, are this kind of sequences always finite for all n?
 * For all i, Gi has at most i vertices.
 * For all i, the degree of every vertex of Gi is at most 3.
 * For all i < j, Gi is not topological minor of Gj.