Certain Classes Of Graphs Research Articles

Exact square coloring of certain classes of graphs: Complexity and algorithms

A vertex coloring of a graph G=(V,E) is called an exact square coloring of G if any two vertices at distance exactly 2 receive different colors. The minimum number of colors required by an exact square coloring is called the exact square chromatic number of G and is denoted by χ[#2](G). Given a graph G and a positive integer k, the Exact Square Coloring Problem is to decide whether G admits an exact square coloring using k colors. It is known that Exact Square Coloring Problem is NP-complete for chordal graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete for undirected path graphs, which is a proper subclass of chordal graphs. However, we give linear time algorithms for computing the exact square chromatic number of proper interval graphs and threshold graphs, which are proper subclasses of chordal graphs. Moreover, for a proper interval graph G, we show that χ[#2](G)≤3. We also propose a polynomial time algorithm to produce an exact square coloring of a block graph G using at most χ[#2](G)+1 colors. Next, we study a lower bound of χ[#2](G). A subset S of vertices of a graph G=(V,E) is called an exact square clique of G if the distance between any two vertices in S is exactly 2. The cardinality of the maximum exact square clique of G is called the exact square clique number of G and is denoted by ω[#2](G). Clearly, ω[#2](G)≤χ[#2](G). Given a graph G and a positive integer k, the problem of deciding whether ω[#2](G) is at least k, is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we strengthen these results by proving that this problem remains NP-complete for undirected path graphs, perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs. We also compute the exact value of ω[#2](G) for proper interval graphs, threshold graphs, block graphs and convex bipartite graphs.

Gorenstein and Cohen–Macaulay matching complexes

Let [Formula: see text] be a simple undirected graph. The family of all matchings of [Formula: see text] forms a simplicial complex called the matching complex of [Formula: see text]. Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of [Formula: see text] is Cohen–Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen–Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen–Macaulay.

Two types of size Ramsey numbers for matchings of small order

For simple graphs G and H, their size Ramsey number is the smallest possible size of F such that for any red-blue coloring of its edges, F contains either a red G or a blue H. Similarly, we can define the connected size Ramsey number by adding the prerequisite that F must be connected. In this paper, we explore the relationships between these size Ramsey numbers and give some results on their values for certain classes of graphs. We are mainly interested in the cases where G is either a 2K 2 or a 3K 2, and where H is either a cycle Cn or a union of paths nPm . Additionally, we improve an upper bound regarding the values of and for certain t and m.

Toric ideals of weighted oriented graphs

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.

Energy of Certain Classes of Graphs Determined by Their Laplacian Degree Product Adjacency Spectrum

In this study, we investigate the Laplacian degree product spectrum and corresponding energy of four families of graphs, namely, complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with a null graph.

On the Merrifield-Simmons index and generating functions for the graphs of the state PL(2,n)

In this study, we compute Merrifield-Simmons index of graphs which corresponds to PL (2, n) by using the decomposition formula. Examination shows that Merri-field-Simmons index of certain classes of graphs deduce as a result in a difference equation. Further, the prominent formula for Merrifield-Simmons index of PL (2, n) and generating function are found as a function of the number n of hexagons in the state of PL (2, n). Also the relations formed for the (PL (2, n, n)) graphs obtained by combining two PL (2, n) with a certain angle between them are given.

On A-energy and S-energy of certain class of graphs

Abstract Let A and S be the adjacency and the Seidel matrix of a graph G respectively. A-energy is the ordinary energy E(G) of a graph G defined as the sum of the absolute values of eigenvalues of A. Analogously, S-energy is the Seidel energy ES(G) of a graph G defined to be the sum of the absolute values of eigenvalues of the Seidel matrix S. In this article, certain class of A-equienergetic and S-equienergetic graphs are presented. Also some linear relations on A-energies and S-energies are given.

Forcing edge detour monophonic number of a graph

‎For a connected graph $G=(V,E)$ of order at least two‎, ‎an edge detour monophonic set of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$‎. ‎The edge detour monophonic number of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$‎. ‎A subset $T$ of $S$ is a forcing edge detour monophonic subset for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$‎. ‎A forcing edge detour monophonic subset for $S$ of minimum cardinality is a minimum forcing edge detour monophonic subset of $S$‎. ‎The forcing edge detour monophonic number $f_{edm}(S)$ in $G$ is the cardinality of a minimum forcing edge detour monophonic subset of $S$‎. ‎The forcing edge detour monophonic number of $G$ is $f_{edm}(G)=min{f_{edm}(S)}$‎, ‎where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$‎. ‎We determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs‎. ‎It is shown that for every pair a‎, ‎b of positive integers with $0leq a<b$ and $bgeq 2$‎, ‎there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$‎.

Optimal Bounds for the k -cut Problem

In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process.

Independent point-set domination in line graphs

Line graph of a graph G is an intersection graph of the edge set E(G) of G. In this paper, we obtain a sharp upper bound on the diameter of graph G whose line graph is an ipsd graph (graph possessing an independent point-set dominating set) by establishing a fundamental relation between diameter of G and diameter of its line graph L(G). We prove that if for a graph G, the length of the longest induced cycle is greater than 5, then its line graph does not possess an ipsd-set. Further we characterize certain classes of graphs viz., trees, complete graphs and complete bipartite graphs whose line graphs possess an independent point set dominating set.

The total edge Steiner number of a graph

A total edge Steiner set of G is an edge Steiner set W such that the subgraph &lt;W&gt; induced by has no isolated vertex. The minimum cardinality of a total edge Steiner set of G is the total edge Steiner number of G and is denoted by ste(G). Some general properties satisfied by this concept are studied. The total edge Steiner numbers of certain classes of graphs is studied. Connected graphs of order p with total edge Steiner number 2 or 3 are characterized. Necessary conditions for total edge Steiner number to be p or p − 1 is given. It is shown that for every pair a and b of integers with 2 ≤ a &lt; b and b &gt; a+ 1, there exists a connected graph G such that se(G) = a and ste(G) = b. Also it shown that for every pair a and b of integers with 4 ≤ a &lt; b and b &gt; a + 1, there exists a connected graph G such that st(G) = a and ste(G) = b.

The forcing total monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a subset T of a minimum total monophonic set S of G is a forcing total monophonic subset for S if S is the unique minimum total monophonic set containing T . A forcing total monophonic subset for S of minimum cardinality is a minimum forcing total monophonic subset of S. The forcing total monophonic number ftm(S) in G is the cardinality of a minimum forcing total monophonic subset of S. The forcing total monophonic number of G is ftm(G) = min{ftm(S)}, where the minimum is taken over all minimum total monophonic sets S in G. We determine bounds for it and find the forcing total monophonic number of certain classes of graphs. It is shown that for every pair a, b of positive integers with 0 ≤ a &lt; b and b ≥ a+4, there exists a connected graph G such that ftm(G) = a and mt(G) = b.

Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

Let $G$ be a graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro were the first to consider a monomial ideal $\mathcal{M}_G$, called the $G$-parking function ideal, in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively to $G$-parking functions. Dochtermann introduced and studied skeleton ideals of the graph $G$, which are subideals of the $G$-parking function ideal with an additional parameter $k ~(0\le k \le n-1)$. A $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of the graph $G$ is generated by monomials corresponding to non-empty subsets of the set of non-root vertices $[n]$ of size at most $k+1$. Dochtermann obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the $k$-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.

Center Smooth Sets and Center Smooth Numbers of Graphs

For any proper set S of V in a graph G(V, E), the S-eccentricity, eG, S (υ) (in short eS (υ)) of a vertex υ in G is . The S-center of G is CS (G) = {υ ∈ V | eS (x) ≤ e S(x)∀x ∈ V} and S 1-eccentricity, eG,S1 (v) (in short eS1 (V)) of a vertex v in S is . S 1-center of G is CS1 (G) = {υ ∈ V | eS1 (x) ≤ e S1 (x)∀x ∈ V}. Then G is called a center-smooth graph if CS (G) = CS1 (G) and the set S is defined to be a center-smooth set. We identify the center smooth sets of certain classes of graphs namely, Kq,p , Kp – q, Kp , wheel graphs and lollipop graph and enumerate them for many of these graph classes. We also introduce the concept of center smooth number, which is defined as the number of distinct center smooth set of a graph G, and determine the center smooth number of some graph classes.

Eccentricity splitting graph of a graph

Let G = (V, E) be any connected graph with for all uj , uk ∈ Si if e(uj ) = e(uk )(1 ≤ i ≤ t) with each | Si |≥ 2 and . The eccentricity splitting graph of a graph denoted by ES(G) is obtained by taking a copy of G and adding vertices w 1, w 2, … , wt such that wi is adjacent only to the vertices of Si for 1 ≤ i ≤ t. We initiate the study on eccentricity splitting graph ES(G) and examine its structural properties. We also analyze diameter, girth and chromatic number of eccentricity splitting graphs of certain classes of graphs.

The monophonic global domination number of a graph

A set M ⊆ V is said to be a monophonic global dominating set of G if M is both a monophonic set and a global dominating set of G. The minimum cardinality of a monophonic global dominating set of G is the monophonic global domination number of G and is denoted by γm(G). A monophonic global dominating set of cardinality γm(G) is called a γm-set of G. The monophonic global domination number of certain classes of graphs are determined. It is proved that 2 ≤ γm(G) ≤ γg (G) ≤ n, where γg (G) is a geodetic global domination number of a G. It is shown that for every pair of positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that γm(G) = a and γg (G) = b.

Harmonious labeling of the union of m-cycle and n-path graph

A graph G is called harmonious graph if there is an injective mapping f from the set of vertices of G to the set Zq so that the mapping induced a bijection mapping f ∗ from the set of edges of G to Zq , which is defined as f*(xy) = f(x)+ f(y) for each edge xy∈ E(G). The mapping f is called harmonious labeling. Harmonious labeling concept has been developed since 1980. At that time, harmonious labeling is inspired by channel assignment problems. Nowadays, harmonious labeling has been known for various classes of graphs. However, there are still many certain classes of graphs that have not been determined yet whether they can be harmonious labeled or not. One of the class of graphs that have not been determined yet whether it can be harminous labeled or not is graph Cm ∪ Pn In this paper, we prove that there is a harmonious labeling of graph Cm ∪ Pn for some values m and n.

Secure Italian domination in graphs

An Italian dominating function (IDF) on a graph G is a function f:V(G)rightarrow {0,1,2} such that for every vertex v with f(v)=0, the total weight of f assigned to the neighbours of v is at least two, i.e., sum _{uin N_G(v)}f(u)ge 2. For any function f:V(G)rightarrow {0,1,2} and any pair of adjacent vertices with f(v) = 0 and u with f(u) > 0, the function f_{urightarrow v} is defined by f_{urightarrow v}(v)=1, f_{urightarrow v}(u)=f(u)-1 and f_{urightarrow v}(x)=f(x) whenever xin V(G){setminus }{u,v}. A secure Italian dominating function on a graph G is defined as an IDF f which satisfies that for every vertex v with f(v)=0, there exists a neighbour u with f(u)>0 such that f_{urightarrow v} is an IDF. The weight of f is omega (f)=sum _{vin V(G) }f(v). The minimum weight among all secure Italian dominating functions on G is the secure Italian domination number of G. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter.

Topological complexity of unordered configuration spaces of certain graphs

The unordered configuration space of n points on a graph Γ, denoted here by UCn(Γ), can be viewed as the space of all configurations of n unlabeled robots on a system of one-dimensional tracks, which is interpreted as a graph Γ. The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by m(Γ). We discuss a combinatorial approach to compute the topological complexity of a “discretized” version of this space, UDn(Γ), and give results for certain classes of graphs. In the first case, we show that for a large class of graphs, as long as the number of robots is at least 2m(Γ), then TC(UDn(Γ))=2m(Γ)+1. In the second, we show that as long as the number of robots is at most half the number of vertex-disjoint cycles in Γ, we have TC(UDn(Γ))=2n+1.

Restrained geodetic domination in graphs

Let [Formula: see text] be a graph with edge set [Formula: see text] and vertex set [Formula: see text]. For a connected graph [Formula: see text], a vertex set [Formula: see text] of [Formula: see text] is said to be a geodetic set if every vertex in [Formula: see text] lies in a shortest path between any pair of vertices in [Formula: see text]. If the geodetic set [Formula: see text] is dominating, then [Formula: see text] is geodetic dominating set. A vertex set [Formula: see text] of [Formula: see text] is said to be a restrained geodetic dominating set if [Formula: see text] is geodetic, dominating and the subgraph induced by [Formula: see text] has no isolated vertex. The minimum cardinality of such set is called restrained geodetic domination (rgd) number. In this paper, rgd number of certain classes of graphs and 2-self-centered graphs was discussed. The restrained geodetic domination is discussed in graph operations such as Cartesian product and join of graphs. Restrained geodetic domination in corona product between a general connected graph and some classes of graphs is also discussed in this paper.