Graphs and Algorithms Exercise 1 (diameter of Graph Complement)
نویسندگان
چکیده
Let F(n,R), where R ⊆ [n], be the family of all forests on the vertex set [n] which consists of exactly |R| trees. Similarly, let T (n,R) be the family of all trees on the vertex set [n+ 1] with the property that NT (n+ 1) = R for every T ∈ T (n,R), i.e. the neighborhood of the vertex labeled with n + 1 is exactly the set of vertices in R. It is easy to see that |F(n,R)| = |T (n,R)| by the following bijection: map a forest F = (V,E) ∈ F(n,R) to a tree T ∈ T (n,R) defined as T = (V ∪ {n+ 1}, E ∪ ⋃
منابع مشابه
Line graphs associated to the maximal graph
Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...
متن کاملSome results on the complement of a new graph associated to a commutative ring
The rings considered in this article are commutative with identity which are not fields. Let R be a ring. A. Alilou, J. Amjadi and Sheikholeslami introduced and investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (0). They called this graph as a new graph as...
متن کاملModelling Decision Problems Via Birkhoff Polyhedra
A compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. Directed Hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. Sets of extrema of Birkhoff polyhedra are mapped to tours ...
متن کاملComplement of Special Chordal Graphs and Vertex Decomposability
In this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially Cohen-Macaulay.
متن کاملDegree distance index of the Mycielskian and its complement
In this paper, we determine the degree distance of the complement of arbitrary Mycielskian graphs. It is well known that almost all graphs have diameter two. We determine this graphical invariant for the Mycielskian of graphs with diameter two.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014