scholarly journals Packing of Mixed Hyperarborescences with Flexible Roots via Matroid Intersection

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Florian Hörsch ◽  
Zoltán Szigeti
Keyword(s):  
Minimum Weight ◽  
Negative Integer ◽  
Matroid Intersection ◽  
Mixed Hypergraph

Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, a non-negative integer $k$ and functions $f,g:V\rightarrow \mathbb{Z}_{\geq 0}$, a packing of $k$ spanning mixed hyperarborescences of $\mathcal{F}$ is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum weight solution to the problem mentioned above.

scholarly journals A Characterization of Gorenstein Rings and Grothendieck's Conjecture

2009 ◽  
Vol 16 (04) ◽  
pp. 653-660
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Regular Sequence ◽  
Negative Integer ◽  
Finitely Generated ◽  
Gorenstein Rings ◽  
Noetherian Local Ring ◽  
Filter Regular Sequence ◽  
System Of Parameters

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x1,…,xd of R which generates an irreducible ideal and [Formula: see text] for all t > 0. Let n be an arbitrary non-negative integer. It is also shown that for an arbitrary ideal 𝔞 of a commutative Noetherian (not necessarily local) ring R and a finitely generated R-module M, [Formula: see text] is finitely generated if and only if there exists an 𝔞-filter regular sequence x1,…,xn∈ 𝔞 such that [Formula: see text] for all t > 0.

Trees with unique Roman dominating functions of minimum weight

2014 ◽  
Vol 06 (03) ◽  
pp. 1450038 ◽  
Author(s):  
Mustapha Chellali ◽  
Nader Jafari Rad
Keyword(s):  
Minimum Weight ◽  
Roman Dominating Function ◽  
Dominating Function

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the value f(V(G)) = ∑u∈V(G)f(u). In this paper we provide a constructive characterization of trees with unique Roman dominating functions of minimum weight.

scholarly journals K-Regular Matroids

2021 ◽  
Author(s):  
◽  
Charles A Semple
Keyword(s):  
Finite Number ◽  
Prime Power ◽  
Negative Integer ◽  
Regular Matroid ◽  
Excluded Minor ◽  
Excluded Minors

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

scholarly journals Strong equality of Roman and perfect Roman domination in trees

2022 ◽  
Author(s):  
Zehui Shao ◽  
Saeed Kosari ◽  
Hadi Rahbani ◽  
Mehdi Sharifzadeh ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

A Roman dominating function (RD-function) on a graph $G = (V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. An Roman dominating function $f$ in a graph $G$ is perfect Roman dominating function (PRD-function) if  every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex  $v$ for which $f(v) = 2$. The (perfect) Roman domination number $\gamma_R(G)$ ($\gamma_{R}^{p}(G)$) is the minimum weight of an (perfect) Roman dominating function on $G$.  We say that $\gamma_{R}^{p}(G)$ strongly equals $\gamma_R(G)$, denoted by $\gamma_{R}^{p}(G)\equiv \gamma_R(G)$, if every RD-function on $G$ of minimum weight is a PRD-function. In this paper we  show that for a given graph $G$, it is NP-hard to decide whether $\gamma_{R}^{p}(G)= \gamma_R(G)$ and also we provide a constructive characterization of trees $T$ with $\gamma_{R}^{p}(T)\equiv \gamma_R(T)$.

scholarly journals Minimum weight resolving sets of grid graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650048
Author(s):  
Patrick Andersen ◽  
Cyriac Grigorious ◽  
Mirka Miller
Keyword(s):  
Shortest Path ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Complete Characterization ◽  
Simple Graph ◽  
Resolving Set ◽  
Maximum Cardinality ◽  
Grid Graphs ◽  
Characterization Of Solutions

For a simple graph [Formula: see text] and for a pair of vertices [Formula: see text], we say that a vertex [Formula: see text] resolves [Formula: see text] and [Formula: see text] if the shortest path from [Formula: see text] to [Formula: see text] is of a different length than the shortest path from [Formula: see text] to [Formula: see text]. A set of vertices [Formula: see text] is a resolving set if for every pair of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a vertex [Formula: see text] that resolves [Formula: see text] and [Formula: see text]. The minimum weight resolving set problem is to find a resolving set [Formula: see text] for a weighted graph [Formula: see text] such that [Formula: see text] is minimum, where [Formula: see text] is the weight of vertex [Formula: see text]. In this paper, we explore the possible solutions of this problem for grid graphs [Formula: see text] where [Formula: see text]. We give a complete characterization of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is [Formula: see text]. We show that the grid has the property that given a landmark set, we only need to investigate whether or not all pairs of vertices that share common neighbors are resolved to determine if the whole graph is resolved. We use this result to provide a characterization of a class of minimals whose cardinalities range from [Formula: see text] to [Formula: see text] and show that the number of such minimals is [Formula: see text].

scholarly journals K-Regular Matroids

2021 ◽  
Author(s):  
◽  
Charles A Semple
Keyword(s):  
Finite Number ◽  
Prime Power ◽  
Negative Integer ◽  
Regular Matroid ◽  
Excluded Minor ◽  
Excluded Minors

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

scholarly journals Vertex-Edge Roman Domination

2021 ◽  
Vol 45 (5) ◽  
pp. 685-698
Author(s):  
H. NARESH KUMAR ◽  
◽  
Y. B. VENKATAKRISHNAN
Keyword(s):  
Lower Bound ◽  
Minimum Weight ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Dominating Function

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number γveR(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γveR(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γveR(T) = 2γ′(T), where γ′(T) is the edge domination number of T.

On trees with domination number equal to edge-vertex roman domination number

2020 ◽  
pp. 2150045
Author(s):  
H. Naresh Kumar ◽  
Y. B. Venkatakrishnan
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

An edge-vertex Roman dominating function (or just ev-RDF) of a graph [Formula: see text] is a function [Formula: see text] such that for each vertex [Formula: see text] either [Formula: see text] where [Formula: see text] is incident with [Formula: see text] or there exists an edge [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text]. The weight of a ev-RDF is the sum of its function values over all edges. The edge-vertex Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of an ev-RDF [Formula: see text]. We provide a characterization of all trees with [Formula: see text], where [Formula: see text] is the domination number of [Formula: see text]

Graphs whose weak Roman domination number increases by the deletion of any edge

2021 ◽  
Author(s):  
Rihab Hamid ◽  
Nour El Houda Bendahib ◽  
Mustapha Chellali ◽  
Nacéra Meddah
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Necessary Condition ◽  
Roman Domination ◽  
Edge Deletion ◽  
Critical Graphs ◽  
Roman Domination Number ◽  
Number Formula ◽  
Roman Dominating Function

Let [Formula: see text] be a function on a graph [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is said to be undefended with respect to [Formula: see text] if it is not adjacent to a vertex [Formula: see text] with [Formula: see text]. A function [Formula: see text] is called a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text], such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], has no undefended vertex. The weight of a WRDF is the sum of its function values over all vertices, and the weak Roman domination number [Formula: see text] is the minimum weight of a WRDF in [Formula: see text]. In this paper, we consider the effects of edge deletion on the weak Roman domination number of a graph. We show that the deletion of an edge of [Formula: see text] can increase the weak Roman domination number by at most 1. Then we give a necessary condition for [Formula: see text]-ER-critical graphs, that is, graphs [Formula: see text] whose weak Roman domination number increases by the deletion of any edge. Restricted to the class of trees, we provide a constructive characterization of all [Formula: see text]-ER-critical trees.

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