A linear time algorithm for minimum equitable dominating set in trees

Let G = (V, E) be a graph. A set D  V (G) is equitable dominating set of G if  v  V – D  a vertex u  D such that uv  E (G) and |d(u) – d(v)|  1. A set D  V (G) is outer equitable dominating set if D is equitable dominating and is connected graph. The outer equitable connected domination number of G is the minimum cardinality of the outer-equitable connected dominating set of G and is denoted by oec(G). We introduce in this paper the concept of total outer equitable connected domination, exact values for some particular classes of graphs are found, some results on total outer equitable domination number are also established.

Considering all physical, biological, and social systems, fuzzy graph (FG) models serve the elemental processes of all natural and artificial structures. As the indeterminate information is an essential real-life problem, which is mostly uncertain, modeling the problems based on FGs is highly demanding for an expert. Vague graphs (VGs) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed in bringing about satisfactory results. In addition, VGs are a very useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. The previous definition restrictions in FGs have made us present new definitions in VGs. A wide range of applications has been attributed to the domination in graph theory for several fields such as facility location problems, school bus routing, modeling biological networks, and coding theory. Concepts from domination also exist in problems involving finding the set of representatives, in monitoring communication and electrical networks, and in land surveying (e.g., minimizing the number of places a surveyor must stand in order to take the height measurement for an entire region). Hence, in this article, we introduce different concepts of dominating, equitable dominating, total equitable dominating, weak (strong) equitable dominating, equitable independent, and perfect dominating sets in VGs and also investigate their properties by some examples. Finally, we present an application in medical sciences to show the importance of domination in VGs.

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

Matchability and [formula omitted]-maximal matchings

In this paper, we consider the Minimum Independent Dominating Set problem and develop exact exponential algorithms that break the trivial O(2|V |) bound. A simple \(O^{*}({\sqrt{3}}^{|V|})\) time algorithm is developed to solve this problem on general graphs. For sparse graphs, e.g. graphs with degree bounded by 3 and 4, we show that a few new branching techniques can be applied to these graphs and the resulting algorithms have time complexities O *(20.465|V |) and O *(20.620|V |), respectively. All our algorithms only need polynomial space.

For a graph G(V,E) with vertex set V and edge set E, a set S ⊆ V is called a power dominating set (PDS), if every vertex u ∈ V - S is observed by some vertices in S using the following rules: (i) if a vertex v in G is in PDS, then it dominates itself and all the adjacent vertices of v and (ii) if an observed vertex v in G has k > 1 adjacent vertices and if k - 1 of these vertices are already observed, then the remaining one non-observed vertex is also observed by v in G. A power dominating set S ⊆ V of G (V, E) is said to be an equitable power dominating set, if for every vertex v ∈ V - S there exists an adjacent vertex u ∈ S such that the difference between degree of u and degree of v is less than or equal to 1, that is |d(u) - d(v)| ≤ 1. The minimum cardinality of an equitable power dominating set of G is called the equitable power domination number of G and is denoted by γepd (G). Let G = (V, E) be a graph with V = S1 ∪ S2 ∪ …St ∪ T where each Si is a set of vertices having cardinality of at least two and the vertices are of the same degree and T = V\ ∪ St. The degree splitting graph of G is obtained from G by adding vertices w1, w2, …, wt and joining wi to each vertex of Si (1 ≤ i ≤ t) and is denoted by DS(G). In this paper we establish the equitable power domination number of the degree splitting graph of certain graphs.

Chordal graphs are triangulated graphs where any cycle larger than three is bisected by a chord. Many combinatorial optimization problems such as computing the size of the maximum clique and the chromatic number are NP-hard on general graphs but have polynomial time solutions on chordal graphs. In this paper, we present a novel multithreaded algorithm to extract a maximal chordal sub graph from a general graph. We develop an iterative approach where each thread can asynchronously update a subset of edges that are dynamically assigned to it per iteration and implement our algorithm on two different multithreaded architectures -- Cray XMT, a massively multithreaded platform, and AMD Magny-Cours, a shared memory multicore platform. In addition to the proof of correctness, we present the performance of our algorithm using a test set of synthetical graphs with up to half-a-billion edges and real world networks from gene correlation studies and demonstrate that our algorithm achieves high scalability for all inputs on both types of architectures.

A linear time algorithm for computing a minimum paired-dominating set of a convex bipartite graph

Let [Formula: see text] and [Formula: see text] be positive integers, and let [Formula: see text] be a simple and connected graph with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text] and let [Formula: see text]. Let [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] locating set of [Formula: see text] if, for any distinct [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] location number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] locating sets of [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] dominating set of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] dominating sets of [Formula: see text]. The [Formula: see text]-location-domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all sets [Formula: see text] such that [Formula: see text] is both a distance-[Formula: see text] locating set and a distance-[Formula: see text] dominating set of [Formula: see text]. For any connected graph [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text]. We characterize connected graphs [Formula: see text] satisfying [Formula: see text] equals [Formula: see text] and [Formula: see text], respectively. We examine the relationship among [Formula: see text], [Formula: see text] and [Formula: see text]; along the way, we show that [Formula: see text] if [Formula: see text]. We also show that there exist graphs [Formula: see text] and [Formula: see text] with [Formula: see text] such that [Formula: see text] can be arbitrarily large. Moreover, we examine some graph classes.

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear time recognition algorithms. However, the problems of testing whether a given (di)graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear time algorithms. While we now know that for any fixed k, we can test in linear time whether a graph is k vertices away from being planar [FOCS 2009, SODA 2014] or bipartite [SODA 2014, SICOMP 2016], the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 78] which runs in time O(nk–1m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] which runs in time O(k!4kk4nm). In other words, it has remained open whether it is possible to recognize in linear time, a graph which is 2 vertices away from being acyclic! In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time O(k!4kk5(n + m)). That is, for every fixed k, our algorithm runs in time O(m + n), thus mirroring the case for planarity and bipartiteness. Our algorithm is designed via a general methodology that shaves off a factor of n from some algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear time algorithms for directed cut-problems and (ii) it can be used in combination with future improvements in algorithms for the compression version of other well-studied cut-problems such as Multicut and Directed Subset Feedback Vertex Set.

The analysis of many processes modeled by directed graphs requires the selection of a subset of vertices which cut all the cycles in the graph. Reducing the size of such a cutset usually leads to a simpler and more efficient analysis, but the problem of finding minimum cutsets in general directed graphs is known to be $NP$-complete. In this paper we show that in reducible graphs (and thus in almost all the “practical” flowcharts of programs), minimum cutsets can be found in linear time. We further show that the linear algorithm can check its own applicability to a given graph, thus eliminating the need of prechecking (in nonlinear time) whether it is reducible or not. An immediate application of this result is in program verification systems based on Floyd’s inductive assertions method.

A set \(S\subseteq V\) of vertices (called codewords) of a graph \(G=(V, E)\) is called a Multiple Intruder Locating Dominating set ( MILD set) if every non-codeword \(v\) is adjacent to at least one codeword \(u\) which is not adjacent to any other non-codeword. This enables one to locate intruders at multiple locations of a network and hence the name. The \(MILD(G)\) is the minimum cardinality of a \(MILD\) set in \(G\). Here, we show that the problem of finding MILD set for general graphs is NP-Complete. Further, we provide a linear time algorithm to find the MILD number of trees through dynamic programming approach and then, we extend the algorithm for unicyclic graphs.

We study subgraph counting over fully dynamic graphs, which undergo edge insertions and deletions. Counting subgraphs is a fundamental problem in graph theory with numerous applications across various fields, including database theory, social network analysis, and computational biology. In database theory, we can use dynamic subgraph counting algorithms on layered graphs to maintain the sizes of joins of databases that undergo updates. Specifically, the problem of finding the number of elements in a cyclic join of size k is equivalent to counting the number of k-cycles in k-layered graphs. For example, let R, S, and T be relations that have schemas (A, B), (B, C), and (C, A) respectively. Then the size of the join of R with S with T is given by the number of triangles in the corresponding layered graph where there is a layer for each attribute, the vertices are the attribute values and the edges represent the tuples of attribute values in the relations. Maintaining the number of triangles in fully dynamic graphs is very well studied and has an upper bound of O(√m) for the update time [KNN+20]. There is also a conditional lower bound of Ω(m 1/2-γ ) for any constant γ>0, for the update time [HKNS15] under the Online Matrix-Vector (OMv) conjecture implying that O(√m) is the ''right answer' for the update time of counting triangles. More recently, [HHH22] studied the problem of maintaining the number of 4-cycles in fully dynamic graphs and designed an algorithm with O(m 2/3 ) update time which is a natural generalization of the approach for counting triangles. They also studied the problem of counting 4-cliques showing that the folklore upper bound of O(m) for the update time is tight under the static combinatorial 4-clique conjecture by giving a lower bound of Ω(m 1-γ ) for any γ>0. Thus, it seems natural that O(m 2/3 ) might be the correct answer for the complexity of the update time for counting 4-cycles. In this work, we present an improved algorithm for maintaining the number of 4-cycles in fully dynamic graphs. Our algorithm achieves a worst-case update time of O(m 2/3-ε ) for some constant ε>0. We also show that the problem of counting 4-cycles is equivalent in layered graphs and general graphs. Our approach crucially uses fast matrix multiplication and leverages recent developments therein to get an improved runtime. Using the current best value of the matrix multiplication exponent ω=2.371339 we get ε=0.009811 and if we assume the best possible exponent i.e. ω=2 then we get ε=1/24. There is also a lower bound of Ω(m 1/2-γ ) for any constant γ>0, for the update time [HKNS15,HHH22], so there is still a big gap between the best-known upper and lower bounds. The key message of our paper is demonstrating that O(m 2/3 ) is not the correct answer for the complexity of the update time.

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.