The problem of bisecting a given angle is counted among planar problems, because, as detailed by Euclid in Elements I.9, to construct the line segment that divides a given angle into two equal parts, we construct (by compass) three circles of equal radius, and then (by straightedge) join the vertex of the angle with the point at which the circles intersect (Euclid 1956, Volume I, 264 265).
The corona product of two graphs X and Y, denoted by (Xbullet Y), is a graph obtained by taking one copy of graph X and (n_{X}) copies of graph Y and joining the vertex X, that is, on the ith position in X to every vertex in ith copy of Y.
The stimulation electrodes were arranged on the skull as follows: the anode was placed at the vertex and the cathode electrodes were positioned 11 12 cm along the line joining the vertex to the acoustic meatus in the parietal region, on each side [ 9, 10].
In a mixed undirected-directed network (henceforth semidirected network), each vertex (person) has an undirected degree representing the number of undirected edges joining the vertex to other vertexes as well as both an in-degree and an out-degree representing the number of directed edges coming from other persons and going to other persons, respectively.
T is any tree on (k+1) vertices; (T=Rcirc k) for some tree R on (ngeq1) vertices, where (Rcirc k) is the graph obtained by taking one copy of R and (|V(R)|) copies of the path (P_{k-1}) of length (k-1) and then joining the ith vertex of R to exactly one end vertex in the ith copy of (P_{k-1}).
Then (gamma_{k}(T =n) if and only if at least one of the following conditions holds: (1) T is any tree on (k+1) vertices; (2) (T=Rcirc k) for some tree R on (ngeq1) vertices, where (Rcirc k) is the graph obtained by taking one copy of R and (|V(R)|) copies of the path (P_{k-1}) of length (k-1) and then joining the ith vertex of R to exactly one end vertex in the ith copy of (P_{k-1}). .
It is obvious that the set (bigcup_{jin J}sigma_{j}) is an arcwise connected subset of S joining the 'upper' vertex (overline{v}^{n}) of S with the 'bottom' (langle{overline{v}^{i}: iin[n-1]} rangle) of S. Since (0in l sigma_{J})), we have that (h_{n} v >0) for the unique vertex of (sigma_{J}) for which (l v =0).
Researchers introduced many graph operations such as the cartesian product, join of graphs, line graphs, the corona product, the edge corona product, the subdivision-vertex join, the subdivision edge join, the neighborhood corona, the subdivision vertex neighborhood corona and the subdivision edge neighborhood corona.
The corona product G 1 ∘ G 2 of two graphs G 1 and G 2 is defined to be the graph Γ obtained by taking one copy of G 1 (which has n 1 vertices) and n 1 copies of G 2, and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2, i = 1, 2, …, n 1.
The corona product G 1 ∘ G 2 of two graphs G 1 and G 2 is defined to be the graph Γ obtained by taking one copy of G 1 (which has | G 1 | vertices) and | G 1 | copies of G 2, and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2, i = 1, 2, …, | G 1 |.
The R-vertex neighborhood corona product of (R(X)) and Y, denoted by (R(X boxdot Y), is a graph obtained from one copy of vertex-disjoint graph (R(X)) and (n_{X}) copies of Y and joining the neighbors of a vertex of X in (R(X)), that is, on the ith position in (R(X)) to every vertex in the ith copy of Y.
