Content may be subject to copyright. # An example of a subgraph G i . The black vertices are included in the set S constructed in the proof of Lemma 11. The cycle with the solid edges is a Hamiltonian cycle of G ′ i , and the dashed lines within the cycle represent edges of G ′ i not in the Hamiltonian cycle. The dotted lines depict a possible partition of the vertices considered in steps 1-3 to obtain an upper bound on |S|.

Source publication
Preprint
Full-text available
We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $\gamma^L_t(G)$. It has...

## Similar publications

Preprint
Full-text available
The fractional differential equation $L^\beta u = f$ posed on a compact metric graph is considered, where $\beta>\frac14$ and $L = \kappa - \frac{\mathrm{d}}{\mathrm{d} x}(H\frac{\mathrm{d}}{\mathrm{d} x})$ is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients $\kappa,H$. We de...
Article
Full-text available
Let t, k, d be integers with 1≤d≤k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le d \le k - 1$$\end{document} and t≥8d+11\documentclass[12pt]{minimal} \usepacka...
Preprint
Full-text available
Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for digraphs with maximum degree $5$ or oriented planar graphs with maximum degree $6$. A digraph is said to be \$...
Preprint
Full-text available
A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A...