Sentence examples for be a cubic graph from inspiring English sources

Let H be a cubic graph on n vertices w 1,…w n.

Let H be a cubic graph on n vertices w 1,…,w n, and let G be the graph defined as in the proof of Theorem 3. Set p=f(n+3)−(n+3).

Every solution of the cubic functional equations is said to be a cubic mapping.

Every solution of the cubic functional equation is said to be a cubic function.

By contrast to the polynomial time algorithms, we show that the seemingly similar edge- (or arc-) deletion problems where we aim for an Eulerian graph are NP-hard, even in the extremely restricted case when the input is a cubic planar graph and the number of deletions can be arbitrary (Theorem 3).

In this section we examine the following problems: The undirected problem can be easily seen to be NP-hard by observing that a cubic graph contains a Hamiltonian cycle if and only if it can be made Eulerian by edge deletions.

We recall that a cubic graph is a 3-regular graph.

See Fig. 2. Fig. 2 The graph G ′ obtained from a cubic graph H. Observe that Δ G ′)=n+3 and the vertices of degree n+3 are v, t 1,…,t p and u n when p=n+1.

Indeed, if deleting a set of edges from a cubic graph G results in an Eulerian graph G′, then each vertex in G′ must have degree 2, so G′ must be a Hamiltonian cycle of G. Since the Hamiltonian Cycle problem restricted to cubic planar graphs is NP-hard [ 16] the result follows.

Fig. 1 Graph G obtained from a cubic graph H.

3-EDGE-COLOURABILITY OF CUBIC GRAPHS Instance: A cubic graph G. Question: Is G 3-edge colourable?