Locating-dominating sets in twin-free graphs

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $\gamma_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] that if $G$ is a twin-free graph of order $n$ without isolated vertices, then $\gamma_L(G)\le \frac{n}{2}$. We prove the general bound $\gamma_L(G)\le \frac{2n}{3}$, slightly improving over the $\lfloor\frac{2n}{3}\rfloor+1$ bound of Garijo et al. We then provide constructions of graphs reaching the $\frac{n}{2}$ bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees $G$ that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.Comment: 11 pages; 4 figure

Microbial Succession in Spontaneously Fermented Grape Must Before, During and After Stuck Fermentation

The microbial succession in spontaneously fermenting Riesling must was investigated from the beginning(pressing) until the end (sulphuring) of the fermentation in two harvest years (2008 and 2009) at a Mosellewinery (Germany). In both years, the fermentation was interrupted by a stuck period. The length of thestuck period varied considerably (20 weeks in 2008 and one week in 2009). Different yeasts (Candida,Debaryomyces, Pichia, Hanseniaspora, Saccharomyces, Metschnikowia, Cryptococcus, Filobasidium andRhodotorula) and bacteria (Gluconobacter, Asaia, Acetobacter, Oenococcus, Lactobacillus, Bacillus andPaenibacillus) were isolated successively by plating. The main fermenting organism was Saccharomycesuvarum. Specific primers were developed for S. uvarum, H. uvarum and C. boidinii, followed by thedetermination of the total cell counts with qPCR. The initial glucose concentration differed between thetwo years and was 116 g/L in 2008 and 85.4 g/L in 2009. Also, the fructose concentrations were differentin both years (114 g/L in 2008 and 77.8 g/L in 2009). The stuck period appeared when the glucose/fructoseratio was 0.34 and 0.12 respectively. The microbiota changed during the stuck period

Remarks about disjoint dominating sets

We solve a number of problems posed by Hedetniemi, Hedetniemi, Laskar, Markus, and Slater concerning pairs of disjoint sets in graphs which are dominating or independent and dominating

An independent dominating set in the complement of a minimum dominating set of a tree

We prove that for every tree $T$ of order at least $2$ and every minimum dominating set $D$ of $T$ which contains at most one endvertex of $T$, there is an independent dominating set $I$ of $T$ which is disjoint from $D$. This confirms a recent conjecture of Johnson, Prier, and Walsh

Partitioning a graph into a dominating set, a total dominating set, and something else

A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, {\it Ars Comb.} {\bf 89} (2008), 159--162) implies that every connected graph of minimum degree at least three has a dominating set $D$ and a total dominating set $T$ which are disjoint. We show that the Petersen graph is the only such graph for which $D\cup T$ necessarily contains all vertices of the graph

Rapid accretion state transitions following the tidal disruption event AT2018fyk

High Energy Astrophysic