scholarly journals Fast Diameter Computation within Split Graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Guillaume Ducoffe ◽  
Michel Habib ◽  
Laurent Viennot
Keyword(s):  
Natural Variation ◽  
Linear Time ◽  
Interval Number ◽  
Stable Set ◽  
Vc Dimension ◽  
Graph Invariants ◽  
Split Graph ◽  
Bounded Treewidth ◽  
Split Graphs ◽  
Stabbing Number

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.

scholarly journals Finding Balance: Split Graphs and Related Classes

10.37236/7091 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Ann N. Trenk
Keyword(s):  
Bipartite Graphs ◽  
Stable Set ◽  
Minimal Set ◽  
Split Graph ◽  
Vertex Set ◽  
Split Graphs

A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the following interesting counting fact: unlabeled, unbalanced split graphs on $n$ vertices can be placed into a bijection with all unlabeled split graphs on $n-1$ or fewer vertices. In this paper we translate these concepts and the theorem to different combinatorial settings: minimal set covers, bipartite graphs with a distinguished block and posets of height one.

scholarly journals Split Graphs: Combinatorial Species and Asymptotics

10.37236/8278 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Justin M. Troyka
Keyword(s):  
Stable Set ◽  
Asymptotic Results ◽  
Split Graph ◽  
Split Graphs ◽  
Almost All

A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to Bína and Přibil (2015), Cheng, Collins, and Trenk (2016), and Collins and Trenk (2018). In both the labeled and unlabeled cases, we give asymptotic results on the number of split graphs, of unbalanced split graphs, and of bicolored graphs, including proving the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are balanced.

Algorithmic Aspects of Outer-Independent Total Roman Domination in Graphs

2021 ◽  
pp. 1-9
Author(s):  
Amit Sharma ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Domination Number ◽  
Chordal Graphs ◽  
Roman Domination ◽  
Induced Subgraph ◽  
Bounded Treewidth ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Domination In Graphs

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

Split Graphs with Specified Dilworth Numbers

1984 ◽  
Vol 27 (1) ◽  
pp. 43-47
Author(s):  
Chiê Nara ◽  
Iwao Sato
Keyword(s):  
Induced Subgraph ◽  
Split Graph ◽  
Complete Part ◽  
Split Graphs

AbstractLet G be a split graph with the independent part IG and the complete part KG. Suppose that the Dilworth number of (IG, ≼) with respect to the vicinal preorder ≼ is two and that of (KG, ≼) is an integer k. We show that G has a specified graph Hk, defined in this paper, as an induced subgraph.

Fast and simple algorithms for counting dominating sets in distance-hereditary graphs

2020 ◽  
pp. 2150012
Author(s):  
Min-Sheng Lin
Keyword(s):  
Linear Time ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Dominating Sets ◽  
Complete Problem ◽  
Split Graphs ◽  
Linear Time Algorithms ◽  
Chordal Bipartite Graphs

Counting dominating sets (DSs) in a graph is a #P-complete problem even for chordal bipartite graphs and split graphs, which are both subclasses of weakly chordal graphs. This paper investigates this problem for distance-hereditary graphs, which is another known subclass of weakly chordal graphs. This work develops linear-time algorithms for counting DSs and their two variants, total DSs and connected DSs in distance-hereditary graphs.

scholarly journals Tractable answer-set programming with weight constraints: bounded treewidth is not enough

2012 ◽  
Vol 14 (2) ◽  
pp. 141-164 ◽  
Author(s):  
REINHARD PICHLER ◽  
STEFAN RÜMMELE ◽  
STEFAN SZEIDER ◽  
STEFAN WOLTRAN
Keyword(s):  
Knowledge Representation ◽  
Linear Time ◽  
Answer Set Programming ◽  
Knowledge Representation And Reasoning ◽  
Cardinality Constraints ◽  
Bounded Treewidth ◽  
Straightforward Application ◽  
Intractable Problems ◽  
Important Extension ◽  
Answer Set

AbstractCardinality constraints or, more generally, weight constraints are well recognized as an important extension of answer-set programming. Clearly, all common algorithmic tasks related to programs with cardinality or weight constraints – like checking the consistency of a program – are intractable. Many intractable problems in the area of knowledge representation and reasoning have been shown to become linear time tractable if the treewidth of the programs or formulas under consideration is bounded by some constant. The goal of this paper is to apply the notion of treewidth to programs with cardinality or weight constraints and to identify tractable fragments. It will turn out that the straightforward application of treewidth to such class of programs does not suffice to obtain tractability. However, by imposing further restrictions, tractability can be achieved.

BOUNDING THE NUMBER OF REDUCED TREES, COGRAPHS, AND SERIES-PARALLEL GRAPHS BY COMPRESSION

2013 ◽  
Vol 05 (02) ◽  
pp. 1360001
Author(s):  
TAKEAKI UNO ◽  
RYUHEI UEHARA ◽  
SHIN-ICHI NAKANO
Keyword(s):  
Upper Bound ◽  
Linear Time ◽  
Point Of View ◽  
Rooted Trees ◽  
Compact Representation ◽  
Theoretical Point ◽  
Practical Point ◽  
Bounded Treewidth ◽  
Ordered Tree ◽  
Parallel Graph

We give an efficient encoding and decoding scheme for computing a compact representation of a graph in one of unordered reduced trees, cographs and series-parallel graphs. The unordered reduced trees are rooted trees in which (i) the ordering of children of each vertex does not matter, and (ii) no vertex has exactly one child. This is one of basic models frequently used in many areas. Our algorithm computes a bit string of length 2ℓ – 1 for a given unordered reduced tree with ℓ ≥ 1 leaves in O(ℓ) time, whereas a known folklore algorithm computes a bit string of length 2n – 2 for an ordered tree with n vertices. Note that in an unordered reduced tree, ℓ ≤ n < 2ℓ holds. To the best of our knowledge this is the first of such a compact representation for unordered reduced trees. From the theoretical point of view, the length of the representation gives us an upper bound of the number of unordered reduced trees with ℓ leaves. Precisely, the number of unordered reduced trees with ℓ leaves is at most 22ℓ–2 for ℓ ≥ 1. Moreover, the encoding and decoding can be done in linear time. Therefore, from the practical point of view, our representation is also useful to store a lot of unordered reduced trees efficiently. We also apply the scheme for computing a compact representation to cographs and series-parallel graphs. We show that each of cographs with n vertices has a compact representation in 2n – 1 bits, and the number of cographs with n vertices is at most 22n–1. The resulting number is close to the number of cographs with n vertices obtained by the enumeration for small n that approximates Cdn/n3/2, where C = 0.4126⋯ and d = 3.5608⋯. Series-parallel graphs are well-investigated in the context of the graphs of bounded treewidth. We give a method to represent a series-parallel graph with m edges in ⌈2.5285m – 2⌉ bits. Hence the number of series-parallel graphs with m edges is at most 2⌈2.5285m–2⌉. As far as the authors know, this is the first non-trivial result about the number of series-parallel graphs. The encoding and decoding of the cographs and series-parallel graphs also can be done in linear time.

scholarly journals Forbidden Subgraphs of Power Graphs

10.37236/9961 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Pallabi Manna ◽  
Peter J. Cameron ◽  
Ranjit Mehatari
Keyword(s):  
Nilpotent Groups ◽  
Chordal Graphs ◽  
Forbidden Subgraphs ◽  
Induced Subgraphs ◽  
Open Problems ◽  
Split Graph ◽  
Forbidden Induced Subgraphs ◽  
Power Graph ◽  
Graph Classes ◽  
Split Graphs

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

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