9,299 research outputs found
Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity
We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable
Fault-Tolerant Detection Systems on the King's Grid
A detection system, modeled in a graph, uses "detectors" on a subset of
vertices to uniquely identify an "intruder" at any vertex. We consider two
types of detection systems: open-locating-dominating (OLD) sets and identifying
codes (ICs). An OLD set gives each vertex a unique, non-empty open neighborhood
of detectors, while an IC provides a unique, non-empty closed neighborhood of
detectors. We explore their fault-tolerant variants: redundant OLD (RED:OLD)
sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most
one detector guarantees the properties of OLD sets and ICs, respectively. This
paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite
king's grid, and presents the proof for the bounds on their minimum densities;
[3/10, 1/3] for RED:OLD sets and [3/11, 1/3] for RED:ICs
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
On two variations of identifying codes
Identifying codes have been introduced in 1998 to model fault-detection in
multiprocessor systems. In this paper, we introduce two variations of
identifying codes: weak codes and light codes. They correspond to
fault-detection by successive rounds. We give exact bounds for those two
definitions for the family of cycles
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