A (simple undirected) graph G = (V,E) with m edges is graceful if it has a distinct vertex labeling f : V −→ {0, 1, 2, 3, . . . ,m} which induces a set of distinct edge labels {|f(u)− f(v)| | uv ∈ E, u, v ∈ V }. The famous Ringel-Kotzig conjecture [9, 15] is that all trees are graceful. The base of a tree T is obtained from T by deleting its one-degree vertices. A caterpillar is a tree whose ba...

In this paper we prove that arbitrary Supersubdivision of disconnected graph is graceful.

Several methods for breaking value symmetries have been proposed recently in the constraint programming community. They can be used in conjunction with variable symmetry breaking methods. However, this combination does not break all symmetries in general. We present a combination of lex constraints and element constraints that can be used to break all combinations of variable and value symmetri...