Perfect Domination Polynomial of Homogeneous Caterpillar Graphs and of Full Binary Trees

Mathematical Notes - Let $$G=(V,E)$$ be a simple graph of order $$n$$ . A set $$S \subseteq V(G)$$ is a perfect dominating set of a graph $$G$$ if every vertex $$v\in V(G)-S$$ is adjacent to...     

On the $$D(2)$$ -Vertex Distinguishing Total Coloring of Graphs with $$\Delta=3$$

Mathematical Notes - A $$D(2)$$ -vertex-distinguishing total coloring of a graph $$G$$ is a proper total coloring such that no pair of vertices, within distance two, has the same set of colors, and...     

$$G$$ -Covering Subgroup Systems for the Class of All $$\sigma$$ -Nilpotent Finite Groups

Mathematical Notes - Let $$\mathfrak F$$ be a nonempty class of groups and let $$G$$ be a finite group. A set $$\Sigma$$ of subgroups of the group $$G$$ is called a $$G$$ -covering subgroup system...     

A Note on Shen’s Conjecture on Groups with Given Same-Order Type

Mathematical Notes - Let $$G$$ be a group. Define an equivalence relation $$\sim$$ on $$G$$ as follows: for $$x,y \in G$$ , $$x \sim y$$ if $$x$$ and $$y$$ have same order. The set of sizes of...     

Characterizations of $$\sigma$$ -Solvable Finite Groups

Mathematical Notes - All the groups considered in this paper are finite, and $$G$$ always denotes a finite group; $$\sigma$$ is a partition of the set $$\mathbb{P}$$ of all primes, i.e.,...     

On Orbit Spaces of Distributive Binary $$G$$ -Spaces

Mathematical Notes - The orbit space of a distributive binary $$G$$ -space is studied. A number of its properties in the case of a compact binarily acting group $$G$$ are established.     

On Intersections of Certain Nilpotent Subgroups in Finite Groups

Mathematical Notes - It is proved that, in any finite group $$G$$ with nilpotent subgroups $$A$$ and $$B$$ and the condition $$A\cap B^g\unlhd\langle A,B^g\rangle$$ for any $$g$$ in $$G$$ ,...     

On Orbits and Bi-invariant Subsets of Binary $$G$$ -Spaces

Mathematical Notes - Orbits and bi-invariant subsets of binary $$G$$ -spaces are studied. The problem of the distributivity of a binary action of a group $$G$$ on a space $$X$$ , which was posed in...     

A generalization of the diameter bound of Liebeck and Shalev for finite simple groups

Acta Mathematica Hungarica - Let $$G$$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant $$c$$ such that whenever $$S$$ is a...     

Projectively Invariant Subgroups of Abelian $$p$$ -Groups

Mathematical Notes - For a projectively invariant subgroup $$C$$ of a reduced $$p$$ -group $$G$$ , a nondecreasing sequence of ordinals and the symbol $$\infty$$ is constructed in which the $$k$$...     

Spectral multipliers on Heisenberg–Reiter and related groups

Annali di Matematica Pura ed Applicata (1923 -) - Let $$L$$ be a homogeneous sublaplacian on a $$2$$ -step stratified Lie group $$G$$ of topological dimension $$d$$ and homogeneous dimension $$Q$$...     

Finite Factorizable Groups with $$\mathbb P$$ -Subnormal $$\mathrm v$$ -Supersolvable and $$\mathrm{sh}$$ -Supersolvable Factors

Mathematical Notes - We study a finite factorized group $$G=AB$$ in the case when the factors $$A$$ and $$B$$ can be connected to $$G$$ by a chain of subgroups with prime indices, and either all...     

On Products of $$\mathrm{F}^*(G)$$ -Subnormal Subgroups of Finite Groups

Mathematical Notes - A subgroup $$H$$ of a finite group $$G$$ is said to be $$\mathrm{F}^*(G)$$ -subnormal if it is subnormal in $$H\mathrm{F}^*(G)$$ , where $$\mathrm{F}^*(G)$$ is the generalized...     

Separation of Coadjoint Orbits of Generalized Diamond Lie Groups

Mathematical Notes - Let $$G$$ be a type I connected and simply connected generalized diamond Lie group defined as the semidirect product of a $$d$$ -dimensional Abelian Lie group $$N$$ with...     

On the Uniqueness Sets of Multiple Walsh Series for Convergence in Cubes

Mathematical Notes - Let $$G^d$$ be a power of the Cantor binary group $$G$$ . The uniqueness problem for a multiple Walsh series on a power of the binary group in the case of convergence in cubes...     

Finite Groups with Formation Subnormal Normalizers of Sylow Subgroups

Mathematical Notes - Let $$\mathfrak{F}$$ be a formation. Properties of the class $$\mathrm{w}^{*}\mathfrak{F}$$ of all groups $$G$$ for which $$\pi(G)\subseteq\pi(\mathfrak{F})$$ and the...     

3-连通3-正则图中的可序子集

图G的k元点集X={x1,x2,…,xk}被称为G的k-可序子集,如果X的任意排列都按序排在G的某个圈上.称G是k-可序图,如果G的每一个k元子集都是G的k-可序子集.称G为k-可序Hamilton图,如果X的任意排列都位于G的Hamilton圈上.研究了3-连通3-正则图的可序子集的存在性问题.     

A POLYNOMIAL ALGORITHM FOR FINDING THE MINIMUM FEEDBACK VERTEX SET OF A 3-REGULAR SIMPLE GRAPH

1IntroductionLetG=(VE)beaconnectedsimplegraph.AsubsetFofvertexsetViscalIedafeedbackvertexsetofGifthegraPhGFisaf0rest.ThecardinaIity0faminimumfeedbackvertexset0fGisdenotedbyf(G).AvertexsubsetJofvertexsetViscaJledallonseparatingindependentsetofG,ifJisanilldependentsetofVandGJisconnected.Thema-xiammcardinalityofnollseparatingindependelltsetofGisden0tedbyz(G)andiscalledthenonseparatingindepelldentnumberofG,AgraphGiscalledacactusifGisc0nnectedandanytwocyclesofGaredisjoint.Avertexvofacon…     

A POLYNOMIAL ALGORITHM FOR FINDING THE MINIMUM FEEDBACK VERTEX SET OF A 3-REGULAR SIMPLE GRAPH 1

A subset of the vertex set of a graph is a feedback vertex set of the graph if the resulting graph is a forest after removed the vertex subset from the graph. A polynomial algorithm for finding a minimum feedback vertex set of a 3-regular simple graph is provided.     

A POLYNOMIAL ALGORITHM FOR FINDING THE MINIMUM FEEDBACKVERTEX SET OF A 3-REGULAR SIMPLE GRAPHR1

A subset of the vertex set of a graph is a feedback vertex set of thegraph if the resulting graph is a forest after removed the vertexsubset from the graph. A polynomial algorithm for finding a minimumfeedback vertex set of a 3-regular simple graph is provided.