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Abstract: An independent coalition in a graph $G$ consists of two disjoint, independent vertex sets $V_1$ and $V_2$, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is an independent dominating set of $G$. An independent coalition partition of $G$ is a partition $\{V_1, \ldots, V_k\}$ of $V(G)$ such that for every $i\in [k]$, either the set $V_i$ consists of a single dominating vertex of $G$, or $V_i$ forms an independent coalition with some other part $V_j$. The independent coalition number $IC(G)$ of $G$ is the maximum order of an independent coalition of $G$. The independent coalition graph ${\rm ICG}(G,\pi)$ of $\pi=\{V_1, \ldots, V_k\}$ (and of $G$) has the vertex set $\{V_1,\ldots, V_k\}$, vertices $V_i$ and $V_j$ being adjacent if $V_i$ and $V_j$ form an independent coalition in $G$. In this paper, a large family of graphs with $IC(G) = 0$ is described and graphs $G$ with $IC(G)\in \{n(G), n(G)-1\}$ characterized. Some properties of ${\rm ICG}(G,\pi)$ are presented. The independent coalition graphs of paths are characterized, and the independent coalition graphs of cycles described.
Keywords: dominating set, independent set, independent coalition, independent coalition number, independent coalition graphs
Published in DiRROS: 11.04.2025; Views: 21; Downloads: 8
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Published in DiRROS: 11.04.2025; Views: 29; Downloads: 17
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Abstract: A subset $S$ of vertices of a graph $G$ is a general position set if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the lower general position number ${\rm gp}^-(G)$ of $G$, which is the number of vertices in a smallest maximal general position set of $G$. We show that ${\rm gp}^-(G) = 2$ if and only if $G$ contains a universal line and determine this number for several classes of graphs, including Kneser graphs $K(n,2)$, line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.
Keywords: general position number, geodetic number, universal line, computational complexity, Kneser graphs, line graphs
Published in DiRROS: 11.04.2025; Views: 26; Downloads: 12
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Keywords: cemented carbide, fracture analysis, stone cutting inserts, chainsaw machine
Published in DiRROS: 11.04.2025; Views: 25; Downloads: 11
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Abstract: The main goal of epidemic modelling is to support the epidemic management through forecasts and analyses of past developments. With this in mind a robust and intuitive SEIR (Susceptible, Exposed, Infectious, Recovered) type model has been developed, applied and verified during the multiple waves of the COVID-19 epidemics in Slovenia since March 2020. The model parameters were based on the general characteristics of the COVID-19 disease reported globally for the entire planet and refined with the aggregate data available mostly on a daily basis in Slovenia, as for example the number of confirmed cases, hospitalized patients, hospitalized patients in intensive care units and deceased. The Slovenian aggregate data was also used to estimate the degree of immunisation due to past infections and vaccination, which reduces the number of susceptible persons for the disease. Examples of the model application are presented to illustrate its robustness and intuitiveness in both the forecasts and analyses of past developments. The analyses of past developments provided specific estimates of modelling parameters for Slovenia and quantified the effects of pharmaceutical and non-pharmaceutical interventions and various events on the development of the epidemics as measured through the reproduction number R. This empirically obtained information was then applied in the forecasts. Accurate forecasts are a great support for decision makers and for hospitals to plan appropriate actions in advance. The inherent uncertainties in the model and data were quantified through intuitive sensitivity analyses represented as different scenarios. The observed accuracy of the forecasts was impressively good also in demanding conditions, when various complex processes influencing the spread of the disease were going on in parallel. This demonstrates the robustness and relevance of the proposed model.
Keywords: epidemic, COVID-19, modelling SEIR, reproduction number
Published in DiRROS: 11.04.2025; Views: 23; Downloads: 12
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