arXiv subjects

Arthur Farley

Publications and source records attributed to Arthur Farley.

2 recordsLinked to original sources

arXiv2026

Totally Disjoint Diametral Paths

In this paper, we study totally disjoint diametral paths in simple connected graphs. A diametral path in a graph is a shortest path that connects two vertices whose mutual distance is equal to the diameter of the graph. Totally disjoint paths are paths that have no vertices in common, including their end vertices. We show that the problem of deciding whether a graph $G$ has $k$ totally disjoint diametral paths is NP-complete. We consider restricted classes of graphs for which the problem of determining the maximum size of a set of totally disjoint diametral paths is readily solved. We then give a linear-time algorithm for a subclass of maximal outerplanar graphs called 2-paths, define a polynomial-time algorithm for threshold graphs, and establish a structural bound for proper interval graphs. Finally, we define classes of extremal graphs with $k$ totally disjoint diametral paths of length $d$ having the fewest possible number of edges.

math.CO↗

arXiv2020

Defensive Domination in Proper Interval Graphs

$k$-defensive domination, a variant of the classical domination problem on graphs, seeks a minimum cardinality vertex set providing a surjective defense against any attack on vertices of cardinality bounded by a parameter $k$. The problem has been shown to be NP-complete} for fixed $k$; if $k$ is part of the input, the problem is not even in NP. We present efficient algorithms solving this problem on proper interval graphs with $k$ part of the input. The algorithms take advantage of the linear orderings of the end points of the intervals associated with vertices to realize a greedy approach to solution. The first algorithm is based on the interval model and has complexity ${\cal O}(n \cdot k)$ for a graph on $n$ vertices. The second one is an improvement of the first and employs bubble representations of proper interval graph to realize an improved complexity of ${\cal O}(n+ \vert{\cal B}\vert \cdot \log k)$ for a graph represented by $\vert{\cal B}\vert$ bubbles.

cs.DM↗