Signed total double Roman dominatıon numbers in digraphs

Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio

Total kk-Rainbow domination numbers in graphs

Let $k\geq 1$ be an integer‎, ‎and let $G$ be a graph‎. ‎A {\it‎ ‎$k$-rainbow dominating function} (or a {\it $k$-RDF}) of $G$ is a‎ ‎function $f$ from the vertex set $V(G)$ to the family of all subsets‎ ‎of $\{1,2,\ldots‎ ,‎k\}$ such that for every $v\in V(G)$ with‎ ‎$f(v)=\emptyset$‎, ‎the condition $\bigcup_{u\in‎ ‎N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled‎, ‎where $N_{G}(v)$ is‎ ‎the open neighborhood of $v$‎. ‎The {\it weight} of a $k$-RDF $f$ of‎ ‎$G$ is the value $\omega (f)=\sum _{v\in V(G)}|f(v)|$‎. ‎A $k$-rainbow‎ ‎dominating function $f$ in a graph with no isolated vertex is called‎ ‎a {\em total $k$-rainbow dominating function} if the subgraph of $G$‎ ‎induced by the set $\{v \in V(G) \mid f (v) \not =\emptyset\}$ has no isolated‎ ‎vertices‎. ‎The {\em total $k$-rainbow domination number} of $G$‎, ‎denoted by‎ ‎$\gamma_{trk}(G)$‎, ‎is the minimum weight of a total $k$-rainbow‎ ‎dominating function on $G$‎. ‎The total $1$-rainbow domination is the‎ ‎same as the total domination‎. ‎In this paper we initiate the‎ ‎study of total $k$-rainbow domination number and we investigate its‎ ‎basic properties‎. ‎In particular‎, ‎we present some sharp bounds on the‎ ‎total $k$-rainbow domination number and we determine the total‎ ‎$k$-rainbow domination number of some classes of graphs‎.

Management of Data and Collaboration for Business Processes

A business process (BP) is a collection of activities and services assembled together to accomplish a business goal. Business process management (BPM) refers to the man- agement and support for a collection of inter-related business processes, which has been playing an essential role in all enterprises. Business practitioners today face enormous difficulties in managing data for BPs due to the fact that the data for BP execution is scattered across databases for enterprise, auxiliary data stores managed by the BPM sys- tems, and even file systems (e.g., definition of BP models). Moreover, current data and business process modeling approaches leave associations of persistent data in databases and data in BPs to the implementation level with little abstraction. Implementing busi- ness logic involves data access from and to database often demands high development efforts.In the current study, we conceptualize the data used in BPs by capturing all needed information for a BP throughout its execution into a “universal artifact”. The concep- tualization provides a foundation for the separation of BP execution and BP data. With the new framework, the data analysis can be carried out without knowing the logic of BPs and the modification of the BP logics can be directly applied without understanding the data structure.Even though universal artifacts provide convenient data access for processes, the data is yet stored in the underlying database and the relationship between data in artifacts and the one in database is still undefined. In general, a way to link the data of these two data sources is needed. we propose a data mapping language aiming to bridge BP data and enterprise database, so that the BP designers only need to focus on business data instead of how to manipulate data by accessing the database. We formulate syntactic conditions upon specified mapping in order that updates upon database or BP data can be properly propagated.In database area, mapping database to a view has been widely studied In recently years, data exchange method extends the notion of database views to a target database (i.e., multiple views) by using a set of conjunctive queries called “tuple generating de- pendency” (tgd). Tgd is a language that is easy to understand/specify, expressive, and decidable for a wide range of properties, which is ideal as a mapping language. Naturally, if both enterprise database and artifacts are represented as relational database, we can take advantage of data exchange technology to bridge enterprise database and artifacts by using tgd as well. Therefore, we re-visit the mapping and update propagation problem under the relational setting.In addition to the data management for a single BP, it is equivalently essential to un- derstand how messages and data should be exchanged among multiple collaborative BPs. With the introduction of artifacts, data is explicitly modeled that can be used in a collab- orative setting. Unfortunately, today’s BP collaboration languages (either orchestration or choreography) do not emphasize on how data is evolved during execution. More- over, the existing languages always assume each participant type has a single participant instance. Therefore, a declarative language is introduced to specify the collaboration among BPs with data and multiple instances concerned. The language adopts a subset of linear temporal logics (LTL) as constraints to restrict the behavior of the collaborative BPs.As a follow-up study, we focus on the satisfiability problem of the declarative BP collaboration language, i.e., whether a given specification as a set of constraints allows at least one finite execution. Naturally, if a specification excludes every possible execution, it should be considered as an undesirable design. Therefore, we consider different combi- nation of the constraint types and for each combination, syntactic conditions are provided to decide whether the given constraints are satisfiable. The syntactic conditions automat- ically lead to polynomial testing methods (comparing to PSPACE-complete complexity of general LTL satisfiability testing)

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs