<p style="text-align: justify;">A caterpillar is a tree obtained from path by attaching pendant vertices. The number of caterpillars size <em>n</em> well-known result. In this work extend result exploring the together with cardinalities stable sets and diameter. Three closed formulas are presented, giving with: (i) smaller set cardinality <em>k</em>, (ii) diameter ...

A graph is shown to be distance-hereditary if and only if no induced subgraph of order five or more has a unique hamiltonian cycle; this is also equivalent to every induced subgraph of order five or more having an even number of hamiltonian cycles. Restricting the induced subgraphs to those of odd order five or more gives two similar characterizations of parity graphs. The close relationship be...

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1, p2 ≥ 1, ..., pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1 , Sp2 , ..., Spd−1 and the path Pd−1 by identifying the root of Spi with the i...

The isotropic matroid M [IAS(G)] of a graph G is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of M [IAS(G)], and if G has at least four vertices,...

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γt(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) ...

Abstract The general position problem is to find the cardinality of largest vertex subset S S such that no triple vertices lies on a common geodesic. For connected graph G G , denoted by gp ( ) {\rm{gp}}\left(G) and called {\rm{...

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1, p2 ≥ 1, ..., pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1 , Sp2 , ..., Spd−1 and the path Pd−1 by identifying the root of Spi with the i...

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by ...

An identifying code $C$ of a graph $G$ is dominating set such that any two distinct vertices have closed neighborhoods within $C$. These codes been widely studied for over decades. We give an improvement all the best known upper bounds, some which stood 20 years, in trees, proving bound $(n+\ell)/2$, where $n$ order and $\ell$ number leaves (pendant vertices) graph. In addition to being size, n...

The teach pendant is a hand-held robot control terminal that provides a convenient means to move the robot, teach locations, and run robot programs. Nowadays, almost teaching pendant is connected with a robot controller using cable. The cable connection and the size of teaching pendant are not problems because a robot controller is separated with robot. However, a large size and wired teaching ...