On the L-Grundy domination number of a graph

In this paper, we continue the study of the L-Grundy domination number of a graph introduced and first studied in [Grundy dominating sequences and zero forcing sets, Discrete Optim. 26 (2017) 66-77]. A vertex in a graph dominates itself and all vertices adjacent to it, while a vertex totally dominates another vertex if they are adjacent. A sequence of distinct vertices in a graph G is called an L-sequence if every vertex v in the sequence is such that v dominates at least one vertex that is not totally dominated by any vertex that precedes v in the sequence. The maximum length of such a sequence is called the L-Grundy domination number, L gr(G), of G. We show that the L-Grundy domination number of every forest G on n vertices equals n, and we provide a linear-time algorithm to find an L-sequence of length n in G. We prove that the decision problem to determine if the L-Grundy domination number of a split graph G is at least k for a given integer k is NP-complete. We establish a lower bound on L gr(G) when G is a regular graph, and investigate graphs G on n vertices for which L gr(G) = n.

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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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A linear time algorithm for minimum equitable dominating set in trees

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-4552 ◽

2021 ◽

Vol 40 (4) ◽

pp. 805-814

Author(s):

Sohel Rana ◽

Sk. Md. Abu Nayeem

Keyword(s):

Linear Time ◽

Dominating Set ◽

Domination Number ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Minimum Cardinality ◽

Np Complete ◽

General Graphs

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

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A Linear Time Algorithm for Embedding Christmas Trees into Certain Trees

Parallel Processing Letters ◽

10.1142/s0129626415500085 ◽

2015 ◽

Vol 25 (04) ◽

pp. 1550008

Author(s):

Indra Rajasingh ◽

R. Sundara Rajan ◽

Paul Manuel

Keyword(s):

Lower Bound ◽

Interconnection Network ◽

Linear Time ◽

Graph Embedding ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Christmas Trees ◽

Host Graph

Graph embedding is an important technique that maps a logical graph into a host graph, usually an interconnection network. In this paper, we compute the exact wirelength of embedding Christmas trees into trees. Moreover, we present an algorithm for embedding Christmas trees into caterpillars with dilation 3 proving that the lower bound obtained in [30] is sharp. Further, we solve the maximum subgraph problem for Christmas trees and provide a linear time algorithm to compute the exact wirelength of embedding Christmas trees into trees.

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Comparing Degenerate Strings

A Mosaic of Computational Topics: from Classical to Novel ◽

10.3233/stal200005 ◽

2020 ◽

Author(s):

Mai Alzamel ◽

Lorraine A.K. Ayad ◽

Giulia Bernardini ◽

Roberto Grossi ◽

Costas S. Iliopoulos ◽

...

Keyword(s):

Lower Bound ◽

Linear Time ◽

Simple Algorithm ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Total Size ◽

Empty Intersection ◽

Compact Representations ◽

String Comparison ◽

Combinatorial Result

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, …, kn-1. Our main result is an O(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in O(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

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Semipaired Domination in Some Subclasses of Chordal Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6782 ◽

2021 ◽

Vol vol. 23 no. 1 (Discrete Algorithms) ◽

Author(s):

Michael A. Henning ◽

Arti Pandey ◽

Vikash Tripathi

Keyword(s):

Linear Time ◽

Dominating Set ◽

Domination Number ◽

Time Algorithm ◽

Chordal Graphs ◽

Minimum Cardinality ◽

Positive Side ◽

Domination Problem ◽

Np Complete ◽

Decision Version

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

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Comparing Degenerate Strings

Fundamenta Informaticae ◽

10.3233/fi-2020-1947 ◽

2020 ◽

Vol 175 (1-4) ◽

pp. 41-58

Author(s):

Mai Alzamel ◽

Lorraine A.K. Ayad ◽

Giulia Bernardini ◽

Roberto Grossi ◽

Costas S. Iliopoulos ◽

...

Keyword(s):

Lower Bound ◽

Linear Time ◽

Simple Algorithm ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Total Size ◽

Empty Intersection ◽

Compact Representations ◽

String Comparison ◽

Combinatorial Result

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

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Proper Orientations of Chordal Graphs

10.5753/etc.2020.11080 ◽

2020 ◽

Author(s):

Julio Araujo ◽

Alexandre Cezar ◽

Carlos Vinícius Gomes Costa Lima ◽

Vinicius Fernandes Dos Santos ◽

Ana Shirley Ferreira Silva

Keyword(s):

Upper Bound ◽

Linear Time ◽

Time Algorithm ◽

Chordal Graphs ◽

Linear Time Algorithm ◽

Threshold Graphs ◽

Minimum Integer ◽

Split Graphs ◽

Np Complete ◽

Proper Orientation

An orientation D of a graph G = (V, E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v ∈ V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u) ≠ dD−(v), for all uv ∈ E(G). An orientation with maximum indegree at most k is called a k-orientation. The proper orientation number of G, denoted by χ→(G), is the minimum integer k such that G admits a proper k-orientation. We prove that determining whether χ→(G) ≤ k is NP-complete for chordal graphs of bounded diameter. We also present a tight upper bound for χ→(G) on split graphs and a linear-time algorithm for quasi-threshold graphs.

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