Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Hammer et al. and Merris, the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure

The principal Erdős–Gallai differences of a degree sequence

The Erdős–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao\u27s order on degree sequences

Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class F of graphs to be dominance monotone if whenever no realization of e contains an element F as an induced subgraph, and d majorizes e, then no realization of d induces an element of F. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3

Linear-Time Recognition of Double-Threshold Graphs

A graph G=(V, E) is a double-threshold graph if there exist a vertex-weight function w:V→ℝ and two real numbers lb, ub ∈ ℝ such that uv ∈ E if and only if 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(n³ m) time, where n and m are the numbers of vertices and edges, respectively

Upward-Closed Hereditary Families In The Dominance Order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class F of graphs to be dominance monotone if whenever no realization of e contains an element F as an induced subgraph, and d majorizes e, then no realization of d induces an element of F. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3

Weakly threshold graphs

We define a weakly threshold sequence to be a degree sequence d = (d1, ⋯, dn) of a graph having the property that Σi≤kdi ≥ k(k - 1) + Σi\u3ek min {k, di} - 1 for all positive k ≤ max {i : di ≥ i - 1}. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs

Weakly threshold graphs

We define a weakly threshold sequence to be a degree sequence $d=(d_1,\dots,d_n)$ of a graph having the property that $\sum_{i \leq k} d_i \geq k(k-1)+\sum_{i > k} \min\{k,d_i\} - 1$ for all positive $k \leq \max\{i:d_i \geq i-1\}$. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs