Minimal Roman Dominating Functions: Extensions and Enumeration

Abstract

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman domination functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph G = (V, E) and a function f : V → {0, 1, 2}, is there a minimal Roman domination function f with f ≤ f ? Here, ≤ lifts 0 < 1 < 2 pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of O(1.9332 n ) for graphs of order n; this is complemented by a lower bound example of Ω(1.7441 n ).

Enumerating Connected Dominating Sets

“…But if Connected Dominating Set Extension is NP-hard, then we cannot expect a polynomial-time algorithm for this more general question, either. On the positive side, it has been recently exemplified with the enumeration problem of minimal Roman domination functions in [2] that a polynomial-time algorithm for the corresponding extension problem can be adapted so that polynomial delay can be achieved for this type of enumeration problem.…”

“…As we also proved that the extension problem associated to CDS is computationally intractable even on 2-degenerate graphs, it is not that straightforward to analyze our enumeration algorithm with the eyes of output-sensitive analysis. Conversely, should it be possible to find an efficient algorithm for an extension problem, also on special graph classes, then usually polynomial-delay algorithms can be shown; as a recent example in the realm of domination problems, we refer to the enumeration of minimal Roman functions described in [2]. So, in the context of our problem, we can ask: Can we achieve polynomial delay for any enumeration algorithm for minimal CDS?…”

Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes

“…To briefly summarize all these findings, in many ways concerning complexity, Roman Domination and Dominating Set behave exactly the same. There are two notable and related exceptions, as delineated in [2], concerning extension problems and output-sensitive enumeration.…”

“…This triggered further interest in looking into enumerating minimal Roman dominating functions on graph classes, as also done in the case of Dominating Set, see [5,20,21,30,32,33]. The basis of the output-sensitive enumeration result of [2] was several combinatorial observations. Here, we find ways how to use the underlying combinatorial ideas for non-trivial enumeration algorithms for minimal Roman dominating functions in split graphs, cobipartite graphs, interval graphs, forests and chordal graphs and for counting these exactly for paths.…”

“…functions) in the next table; new results are shown with boxes; for matching bounds, only one number is displayed; c.f. [2,21,27,33]. Polynomial delay is achievable for the mentioned special graph classes for enumerating minimal dominating sets [36,38], but it is unclear how to combine these…”