Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

Shortest paths between shortest paths and independent sets

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships between these models, observing that shortest path reconfiguration is a special case of independent set reconfiguration in perfect graphs, under any of the three models. Finally, we give polynomial results for restricted classes of graphs (even-hole-free and $P_4$-free graphs)

Domination parameters with number 2: interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number, $\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number, $\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times 2}(G)$, where $G$ is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, $\gamma_R(G)$, and two classical parameters, the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure

Complexity Results for Equistable Graphs and Related Classes

The class of equistable graphs is defined by the existence of a cost structure on the vertices such that the maximal stable sets are characterized by their costs. This graph class, not contained in any nontrivial hereditary class, has so far been studied mostly from a structural point of view; characterizations and polynomial time recognition algorithms have been obtained for special cases. We focus on complexity issues for equistable graphs and related classes. We describe a simple pseudo-polynomial-time dynamic programming algorithm to solve the maximum weight stable set problem along with the weighted independent domination problem in some classes of graphs, including equistable graphs. Our results are obtained within the wider context of Boolean optimization; corresponding hardness results are also provided. More specifically, we show that the above problems are APX-hard for equistable graphs and that it is co-NP-complete to determine whether a given cost function on the vertices of a graph defines an equistable cost structure of that graph.Germany. Federal Ministry of Education and ResearchAlexander von Humboldt-Stiftung (Sofja Kovalevskaja Award 2004

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2

Rapidly evolving Creutzfeldt-Jakob disease in COVID-19: from early status epilepticus to fatal outcome

We report the case of a 70-year-old man coming to our attention for new onset refractory status epilepticus (NORSE) in a rapidly evolving CJD during SARS-CoV-2 co-infection. Our case report describes a fulminant CJD evolution associated with SARS-CoV-2 infection, which led to patient death after 15 days from admission. First EEG presented continuous diffuse spikes, sharp waves and sharp-and-slow wave complexes, pattern consistent with a non-convulsive status epilepticus (NORSE). Our case supports how CJD with SARS-CoV-2 co-infection could be characterized by an accelerated evolution, as already hypothesize for others microorganism infections, and how the diagnosis might be more challenging due to its uncommon presentations, such as NORSE