Treelike Comparability Graphs

Let $S(G^{\sigma})$ be the skew-adjacency matrix of the oriented graph $G^{\sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $\sigma$ to each of its edges. The skew energy of an oriented graph $G^{\sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{\sigma})$. Two oriented graphs are said to be skew equienergetic if their skew energies are equal. In this paper, we determine the skew spectra of some new oriented graphs. As applications, we give some new methods to construct new non-cospectral skew equienergetic oriented graphs.

In this paper we show that the recognition problem for C-I graphs of posets is NP-complete. On the other hand, we prove that induced subgraphs of C-I graphs are exactly complements of comparability graphs, and hence the recognition problem for induced subgraphs of C-I graphs of posets is polynomial.

On edge transitivity of directed graphs

The solution of many ventilation problems involving networks (such as finding routes of minimum length, construction of a network of sections, determination of the cross sections of the initial network with minimum transmission capacity, and estimation of the quality of the initial information obtained in the production of depression records) is much simplified if, instead of sets of all routes and cross sections of the original network, we calculate with onl~ a subset of independent routes and cross sections of the same network [i, 2]. Existing graphical methods of constructing (cross) sections of the original network can mainly be used only to find sections of simple networks [2]. There are no effective methods of constructing independent routes of the original network. By the methods given below, one can relatively easily construct all the independent routes and sections for an Oriented graph (network) G = (X, U, M) with any number of branches, where X = {x,, xl, .... xl, xj, .... Xn} is the set of vertices (nodes), U = {uij} is the set of branches (arcs), and M = l~} is the set of paths in the network G from the -'~ntry"x, to the "exit" x n.

Computing K-terminal reliability of d-trapezoid graphs

Comparability graph augmentation for some multiprocessor scheduling problems

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

A simple graph is reflexive if its second largest eigenvalue ?2 is less than or equal to 2. A graph is a cactus, or a treelike graph, if any pair of its cycles (circuits) has at most one common vertex. For a lot of cactuses the property ?2 ? 2 can be tested by identifying and deleting a single cut-vetex (Theorem 1). if this theorem cannot be applied to a connected reflexive cactus and if all its cycles do not form a bundle, such a graph has at most five cycles. On the same conditions, in this paper we find some classes of maximal reflexive cactuses with four cycles. The complete case of four cycles, together with that of five cycles, is being settled in [10].

The maximum independent set problem (MaxIS) on general graphs is known to be NP-hard to approximate within a factor of $n^{1-epsilon}$, for any $epsilon > 0$. However, there are many ``easy classes of graphs on which the problem can be solved in polynomial time. In this context, an interesting question is that of computing the maximum independent set in a graph that can be expressed as the union of a small number of graphs from an easy class. The MaxIS problem has been studied on unions of interval graphs and chordal graphs. We study the MaxIS problem on unions of perfect graphs (which generalize the above two classes). We present an $O(sqrt{n})$-approximation algorithm when the input graph is the union of two perfect graphs. We also show that the MaxIS problem on unions of two comparability graphs (a subclass of perfect graphs) cannot be approximated within any constant factor.

Let G = (V, E) be a simple graph with n vertices, and let A be an abelian group of order n. Let f: V(G) → A be a bijection, and define the weight of a vertex x by w(x) = Σy ∈ N(x) f(y), where N(x) denotes the open neighborhood of x. We call f an A-distance magic labeling if all vertex weights are equal, and an A-distance antimagic labeling if the weights of distinct vertices are pairwise different. The prime graph of a commutative ring R is the simple graph with vertex set R in which two vertices x and y are adjacent if and only if xRy = {0}. In this paper, we investigate the existence of A-distance magic and A-distance antimagic labelings of the prime graph over the ring Zn for several values of n.

The noncommuting graph $nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs (G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi |i = 1,2,⋯m}∪{vij |i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi ). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs (G ∘ H) and the exact values of χs (G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs (G ∘ H) ≤ χs (G) + χs (H); χs (Pm ∘ H) = χs (H) + 2; χs (K 1,m ∘ H) = χs (H) + 2; χs (Cn ∘ H) = χs (H) + 2, where n ≠ 5.

Let G be a simple graph with vertex set {v1,v2,...,vn} .T he common neighborhood graph (congraph) of G, denoted by con(G), is the graph with vertex set {v1,v2,...,vn}, in which two vertices are adjacent if and only they have at least one common neighbor in the graph G.T he basic properties of con(G) and of its energy are established.

The last chapter of this book eventually presents some famous classes of perfect graphs, namely comparability, interval, permutation, and split graphs. We first present an example of a graph which is at the same time a triangulated, a comparability, a split, and a permutation graph. The importance to be an interval is illustrated with Berge’s mystery story. We discuss furthermore the generation of permutation graphs and close with how to recognise that a given graph is in fact a permutation graph.

An implicit representation of chordal comparability graphs in linear time