We show that, for any n-vertex graph G and integer parameter k, there are at most 34k−n4n−3k maximal independent sets I ⊂ G with |I| ≤ k, and that all such sets can be listed in time O(34k−n4n−3k). These bounds are tight when n/4 ≤ k ≤ n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 3/4)) ≈ 2.4150, improving a previous O((1 + 3)) ≈ 2.4422 alg...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γtr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then γtr(T ) ≥ d(n + 2)/2e. Moreover, we s...

In this paper, we consider graphs having a unique minimum independent dominating set. We first discuss the effects of deleting a vertex, or the closed neighborhood of a vertex, from such graphs. We then discuss five operations which, in certain circumstances, can be used to combine two graphs, each having a unique minimum independent dominating set, to produce a new graph also having a unique m...

Teichmüller’s classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given point of the domain to another given point of the domain. For a domain D ⊂ Rn , n ≥ 2 , we consider the class of all Kquasiconformal maps of D onto itself with...

A graph G is 3-domination-critical (3-critical, for short), if its domination number γ is 3 and the addition of any edge decreases γ by 1. In this paper, we show that every 3-critical graph with independence number 4 and minimum degree 3 is Hamilton-connected. Combining the result with those in [2], [4] and [5], we solve the following conjecture: a connected 3critical graph G is Hamilton-connec...

Independent sets play an important role in matroid theory. In this paper, the definitions of pre-independent fuzzy set system and independent fuzzy set system in L-fuzzy setting are presented. Independent M-fuzzifying set system is introduced and some of its properties are discussed. Further independent (L,M)-fuzzy set system is given and some of its properties are obtained. The relations of th...

For a graphical property P and a graph G, we say that a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. Then the P-domination number of G is the minimum cardinality of a dominating P-set and the P-independence number the maximum cardinality of a P-set. We show that several properties of domination, independent domination and acyclic domination hold for ...

The following inequality chain has been extensively studied in the discrete mathematical literature: i r ~ y ~ i ~ f l ~ F ~IR, where ir and IR denote the lower and upper irredundance numbers of a graph, 2: and F denote the lower and upper domination numbers of a graph, i denotes the independent domination number and fl denotes the vertex independence number of a graph. More than one hundred pa...

The paper describes the hybrid methods of mammogram recognition which are based on independent component analysis, principal component analysis and rough set theory.