A graph with degree sequence $\pi$ is a \emph{unigraph} if it is isomorphic to every graph that has degree sequence $\pi$. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary closure of unigraphs, consisting of all graphs induced in a unigraph. We characterize the class HCU in multiple ways making use of the tools of a decomposition due to Tyshkevich and a partial order on degree sequences due to Rao. We also provide a new characterization of the class that consists of unigraphs for which all induced subgraphs are also unigraphs.

In this paper, we study split graphs and related classes of graphs from the perspective of their sequence of vertex degrees and an associated lattice under majorization. Following the work of Merris [16], we define blocks [α(π)|β(π)], where π is the degree sequence of a graph, and α(π) and β(π) are sequences arising from π. We use the block representation [α(π)|β(π)] to characterize membership in each of the following classes: unbalanced split graphs, balanced split graphs, pseudo-split graphs, and three kinds of Nordhaus-Gaddum graphs (defined in [5], [3]). As in [16], we form a poset under the relation majorization in which the elements are the blocks [α(π)|β(π)] representing split graphs with a fixed number of edges. We partition this poset in several interesting ways using what we call amphoras, and prove upward and downward closure results for blocks arising from different families of graphs. Finally, we show that the poset becomes a lattice when a maximum and minimum element are added, and we prove properties of the meet and join of two blocks.

The product power throttling number of a graph is defined to study prod-uct throttling for power domination. The domination number of a graph is an upper bound for its product power throttling number. It is estab-lished that the two parameters are equal for certain families including paths, cycles, complete graphs, unit interval graphs, and grid graphs (on the plane, cylinder, and torus). Families of graphs for which the prod-uct power throttling number is less than the domination number are also exhibited. Graphs with extremely high or low product power throttling number are characterized and bounds on the product power throttling number are established.

Rabinovitch showed in 1978 that the interval orders having a representation consisting of only closed unit intervals have order dimension at most 3. This article shows that the same dimension bound applies to two other classes of posets: those having a representation consisting of unit intervals (but with a mixture of open and closed intervals allowed) and those having a representation consisting of closed intervals with lengths in {0,1}.

In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice L and its set Q(L) of join-irreducibles, we use classic lattice theory to show that any linear extension of Q(L) generates a distinguishing 2-coloring of L. We prove general upper bounds for the distinguishing chromatic number and particular upper bounds for the Boolean lattice and for divisibility lattices. In addition, we show that the distinguishing number of any twin-free Cohen-Macaulay planar lattice is at most 2.

Throttling addresses the question of minimizing the sum or the product of the resources used in a graph searching process and the time needed to complete the process. The study of throttling began with the study of sum throttling, and parameters that have been studied include various types of zero forcing, power domination, and cops and robbers. Recently two different definitions of product throttling have been introduced for cops and robbers and power domination. This chapter summarizes prior results for these two cases and introduces universal versions of the two definitions. Each of the definitions is then applied to each of the following parameters: standard zero forcing, positive semidefinite zero forcing, power domination, and cops and robbers.

This chapter provides an introduction to split graphs and related classes, presenting both classical results and recent advances. Degree sequences play a crucial role, and we study these geometrically using Ferrers diagrams in Section 4, and as the basis for three-part partitions in Section 5. Bijections between graph classes and a formula for counting the number of unlabelled split graphs on n vertices allow us to count additional classes of graphs related to split graphs. We conclude by presenting a graph decomposition theorem in which split graphs play a starring role.

Throttling addresses the question of minimizing the sum or the product of the resources used to accomplish a task and the time needed to complete that task for various graph searching processes. Graph parameters of interest include various types of zero forcing, power domination, and Cops and Robbers. We provide a survey of product throttling for these parameters.

A poset P=(X,≺) has an interval representation if each x∈X can be assigned a real interval Ix so that x≺y in P if and only if Ix lies completely to the left of Iy. Such orders are called interval orders. In this paper we give a surprisingly simple forbidden poset characterization of those posets that have an interval representation in which each interval length is either 0 or 1. In addition, for posets (X,≺) with a weight of 1 or 2 assigned to each point, we characterize those that have an interval representation in which for each x∈X the length of the interval assigned to x equals the weight assigned to x. For both problems we can determine in polynomial time whether the desired interval representation is possible and in the affirmative case, produce such a representation.

A poset $P= (X, \prec)$ has an interval representation if each $x \in X$ can be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \emph{interval orders}. Fishburn proved that for any positive integer $k$, an interval order has a representation in which all interval lengths are between $1$ and $k$ if and only if the order does not contain $\mathbf{(k+2)+1}$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.

A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the following interesting counting fact: unlabeled, unbalanced split graphs on $n$ vertices can be placed into a bijection with all unlabeled split graphs on $n-1$ or fewer vertices. In this paper we translate these concepts and the theorem to different combinatorial settings: minimal set covers, bipartite graphs with a distinguished block and posets of height one.

A poset P = (X, ≺) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ≺ y in P precisely when each point of I (x) is less than each point in I (y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.

A graph G is an NG-graph if χ(G)+χ(G¯)=|V(G)|+1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also explore the connections between NG-graphs and split graphs. There are three types of NG-graphs and split graphs can also be divided naturally into two categories, balanced and unbalanced. We characterize each of these five classes by degree sequence. We construct bijections between classes of NG-graphs and balanced and unbalanced split graphs which, together with the known formula for the number of split graphs on n vertices, allows us to compute the sizes of each of these classes. Finally, we provide a bijection between unbalanced split graphs on n vertices and split graphs on n−1 or fewer vertices providing evidence for our conjecture that the rapid growth in the number of split graphs comes from the balanced split graphs.

In this paper we extend the work of Rautenbach and Szwarcfiter by giving a structural characterization of graphs that can be represented by the intersection of unit intervals that may or may not contain their endpoints. A characterization was proved independently by Joos, however our approach provides an algorithm that produces such a representation, as well as a forbidden graph characterization.

Nordhaus and Gaddum proved, for any graph $G$, that $\chi(G) + \chi(\overline{G}) \leq n + 1$, where $\chi$ is the chromatic number and $n=|V(G)|$. Finck characterized the class of graphs, which we call NG-graphs, that satisfy equality in this bound. In this paper, we provide a new characterization of NG-graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of the chromatic number of NG-graphs. Our motivation comes from our theorem that generalizes the Nordhaus-Gaddum theorem to the distinguishing chromatic number. For any graph $G$, $\chi_D(G) +\chi_D(\overline{G})\leq n+D(G)$. We call the set of graphs that satisfy equality in this bound NGD-graphs, and characterize the set of graphs that are simultaneously NG-graphs and NGD-graphs.

Linear discrepancy and weak discrepancy have been studied as a measure of fairness in giving integer ranks to the points of a poset. In linear discrepancy, the points are totally ordered, while in weak discrepancy, ties in rank are permitted. In this paper we study the t-discrepancy of a poset, which can be viewed as a hybrid between linear and weak discrepancy, in which at most t points can receive the same rank. Interestingly, t-discrepancy is not a comparability invariant while both linear and weak discrepancy are. We show that for a poset P and positive integers t and k, the decision problem of determining whether the t-discrepancy of P is at most k is NP-complete in general; however, we give a polynomial time algorithm for computing the t-discrepancy of a semiorder. We also find the t-discrepancy for posets that are the disjoint sum of chains.

The fractional weak discrepancy wdF(P) of a poset P = (V, (sic)) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative k for which there exists a function f : V -> R satisfying (i) if a (sic) b then f (a) + 1 <= f (b) and (ii) if a parallel to b then vertical bar f(a)-f(b)vertical bar <= k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5,7] on the range of wd(F) for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0 <= s - 1 <= r < 2s. (C) 2010 Elsevier B.V. All rights reserved.

In this paper we introduce the notion of the total linear discrepancy of a poset as a way of measuring the fairness of linear extensions. If L is a linear extension of a poset P, and x,y is an incomparable pair in P, the height difference between x and y in L is |L(x)−L(y)|. The total linear discrepancy of P in L is the sum over all incomparable pairs of these height differences. The total linear discrepancy of P is the minimum of this sum taken over all linear extensions L of P. While the problem of computing the (ordinary) linear discrepancy of a poset is NP-complete, the total linear discrepancy can be computed in polynomial time. Indeed, in this paper, we characterize those linear extensions that are optimal for total linear discrepancy. The characterization provides an easy way to count the number of optimal linear extensions.

Collins and Trenk define the distinguishing chromatic number $\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity. They prove results about $\chi_D(G)$ based on the underlying graph $G$. In this paper we prove results that relate $\chi_D(G)$ to the automorphism group of $G$. We prove two upper bounds for $\chi_D(G)$ in terms of the chromatic number $\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \cdots p_k^{i_k}$ then $\chi_D(G) \le \chi(G) + i_1 + i_2 \cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\Bbb Z}_{p_{1}^{i_{1}}}\times \cdots \times {\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\chi_D(G) \le \chi(G) + k$. In addition, we characterize automorphism groups of graphs with $\chi_D(G) = 2$ and discuss similar results for graphs with $\chi_D(G)=3$.