Linear-Time Recognition of Double-Threshold Graphs

A graph $$G = (V,E)$$ G = ( V , E ) is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$ w : V → R and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$ lb , ub ∈ R such that $$uv \in E$$ u v ∈ E if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$ lb ≤ w ( u ) + w ( v ) ≤ ub . In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$ O ( n 3 m ) time, where n and m are the numbers of vertices and edges, respectively.

Modular Irregular Labeling on Double-Star and Friendship Graphs

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion

All graph in this paper are simple, finite, and connected. Let be a labeling of a graph . The function is called antimagic rainbow edge labeling if for any two vertices and , all internal vertices in path have different weight, where the weight of vertex is the sum of its incident edges label. The vertex weight denoted by for every . If G has a antimagic rainbow edge labeling, then is a antimagic rainbow vertex connection, where the every vertex is assigned with the color . The antimagic rainbow vertex connection number of , denoted by , is the minimum colors taken over all rainbow vertex connection induced by antimagic rainbow edge labeling of . In this paper, we determined the exact value of the antimagic rainbow vertex connection number of path ( ), wheel ( ), friendship ( ), and fan ( ).

Local Antimagic Chromatic Number for Copies of Graphs

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

Weighted k-domination problem in fuzzy networks

In real-life scenarios, both the vertex weight and edge weight in a network are hard to define exactly. We can incorporate the fuzziness into a network to handle this type of uncertain situation. Here, we use triangular fuzzy number to describe the vertex weight and edge weight of a fuzzy network G. In this paper, we consider weighted k-domination problem in fuzzy networks. The weighted k-domination (WKD) problem is to find a k dominating set D which minimizes the cost $f(D):=\sum_{u\in D}w(u)+\sum_{v\in V\setminus D}\min\{\sum_{u\in S}w(uv)|S\subseteq N(v)\cap D, |S|=k\}$. First, we put forward an integer linear programming model with a polynomial number of constrains for the WKD problem. If G is a cycle, we design a dynamic algorithm to determine its exact weighted 2-domination number. If G is a tree, we give a label algorithm to determine its exact weighted 2-domination number. Combining a primal-dual method and a greedy method, we put forward an approximation algorithm for general fuzzy network on the WKD problem. Finally, we describe an application of the WKD problem to police camp problems.2010 Mathematics Subject Classification. 05C69, 05C35, 03E72

The total disjoint irregularity strength of some certain graphs

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Under a totally irregular total </span><em>k</em><span>-labeling of a graph </span><span><em>G</em> </span><span>= (</span><span><em>V</em>,<em>E</em></span><span>), we found that for some certain graphs, the edge-weight set </span><em>W</em><span>(</span><em>E</em><span>) and the vertex-weight set </span><em>W</em><span>(</span><em>V</em><span>) of </span><span><em>G</em> </span><span>which are induced by </span><span><em>k</em> </span><span>= </span><span>ts</span><span>(</span><em>G</em><span>), </span><em>W</em><span>(</span><em>E</em><span>) </span><span>∩ </span><em>W</em><span>(</span><em>V</em><span>) is a non empty set. For which </span><span>k</span><span>, a graph </span><span>G </span><span>has a totally irregular total labeling if </span><em>W</em><span>(</span><em>E</em><span>) </span><span>∩ </span><em>W</em><span>(</span><em>V</em><span>) = </span><span>∅</span><span>? We introduce the total disjoint irregularity strength, denoted by </span><span>ds</span><span>(</span><em>G</em><span>), as the minimum value </span><span><em>k</em> </span><span>where this condition satisfied. We provide the lower bound of </span><span>ds</span><span>(</span><em>G</em><span>) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.</span></p></div></div></div>

On the Total Vertex Irregular Labeling of Proper Interval Graphs

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V, E) with vertex set V and edge set E, a labeling Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

On the total vertex irregularity strength of comb product of two cycles and two stars

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a graph and <em>k</em> be a positive integer. A total <em>k</em>-labeling of <em>G</em> is a map <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,<em>k</em>}. The vertex weight <em>v</em> under the labeling <em>f</em> is denoted by w_<em>f</em>(<em>v</em>) and defined by <em>w</em>_<em>f</em>(<em>v</em>) = <em>f</em>(<em>v</em>) + \sum_{uv \in{E(G)}} {<em>f</em>(<em>uv</em>)}. A total <em>k</em>-labeling of <em>G</em> is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of <em>G</em>, denoted by <em>tvs</em>(<em>G</em>), is the minimum <em>k</em> such that <em>G</em> has a vertex irregular total <em>k</em>-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let <em>G</em> and <em>H</em> be two connected graphs. Let <em>o</em> be a vertex of <em>H</em>. The comb product between <em>G</em> and <em>H</em>, denoted by <em>G </em>\rhd_o <em>H</em>, is a graph obtained by taking one copy of <em>G</em> and |<em>V</em>(<em>G</em>)| copies of <em>H</em> and grafting the i-th copy of <em>H</em> at the vertex <em>o</em> to the i-th vertex of <em>G</em>. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

On The Total Edge and Vertex Irregularity Strength of Some Graphs Obtained from Star

Let $G=(V(G),E(G))$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ is a map $f: V(G)\cup E(G)\rightarrow \{1,2,\ldots,k \}$. The edge weight $uv$ under the labeling $f$ is denoted by $w_f(uv)$ and defined by $w_f(uv)=f(u)+f(uv)+f(v)$. The vertex weight $v$ under the labeling $f$ is denoted by $w_f(v)$ and defined by $w_f(v) = f(v) + \sum_{uv \in{E(G)}} {f(uv)}$. A total $k$-labeling of $G$ is called an edge irregular total $k$-labeling of $G$ if $w_f(e_1)\neq w_f(e_2)$ for every two distinct edges $e_1$ and $e_2$ in $E(G)$. The total edge irregularity strength of $G$, denoted by $tes(G)$, is the minimum $k$ for which $G$ has an edge irregular total $k$-labeling. A total $k$-labeling of $G$ is called a vertex irregular total $k$-labeling of $G$ if $w_f(v_1)\neq w_f(v_2)$ for every two distinct vertices $v_1$ and $v_2$ in $V(G)$. The total vertex irregularity strength of $G$, denoted by $tvs(G)$, is the minimum $k$ for which $G$ has a vertex irregular total $k$-labeling. In this paper, we determine the total edge irregularity strength and the total vertex irregularity strength of some graphs obtained from star, which are gear, fungus, and some copies of stars.