Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations r-IC and r-LD where the closed neighborhood is considered up to distance r. It also contains Metric Dimension (MD) and its refinement r-MD in which the distance between two vertices is considered as infinite if the real distance exceeds r. Note that while IC = 1-IC and LD = 1-LD, we have MD = $$\infty$$ -MD; we say that MD is not local. In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We focus on two families of problems from the meta-problem: the first one, called local, contains r-IC, r-LD and r-MD for each positive integer r while the second one, called 1-layered, contains LD, MD and r-MD for each positive integer r. We have: (1) the 1-layered problems are NP-hard even in bipartite apex graphs, (2) the local problems are NP-hard even in bipartite planar graphs, (3) assuming ETH, all these problems cannot be solved in $$2^{o(\sqrt{n})}$$ when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in $$2^{o(n)}$$ on bipartite graphs, and (4) except if $${\mathsf{W}[0]} = {\mathsf{W}[2]}$$ , they do not admit parameterized algorithms in $$2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$$ even when restricted to bipartite graphs. Here k is the solution size of a relevant identifying set. In particular, Metric Dimension cannot be solved in $$2^{o(n)}$$ under ETH, answering a question of Hartung and Nichterlein (Proceedings of the 28th conference on computational complexity, CCC, 2013).

A simple digraph is semicomplete if for any two of its vertices u and v , at least one of the arcs ( u , v ) and ( v , u ) is present. We study the complexity of computing two layout parameters of semicomplete digraphs: cutwidth and optimal linear arrangement (O la ). We prove the following: • Both parameters are NP-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis. • The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polynomial kernel unless NP ⊆ coNP/poly. By contrast, O la admits a linear kernel. These results essentially complete the complexity analysis of computing cutwidth and O la on semicomplete digraphs (with respect to standard parameters). Our techniques also can be used to analyze the sizes of minimal obstructions for having a small cutwidth under the induced subdigraph relation.

Let c, k be two positive integers. Given a graph $$G=(V,E)$$ , the c-Load Coloring problem asks whether there is a c-coloring $$\varphi : V \rightarrow [c]$$ such that for every $$i \in [c]$$ , there are at least k edges with both endvertices colored i. Gutin and Jones (Inf Process Lett 114:446–449, 2014) studied this problem with $$c=2$$ . They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of $$c=2$$ , we obtain a kernel with less than 4k vertices and less than $$6k+(3+\sqrt{2})\sqrt{k}+4$$ edges. These results imply that for any fixed $$c\ge 2$$ , c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.