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Let $t$ be a positive real number. A graph is called $t$-tough, if the removal of any cutset $S$ leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough, if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. In this paper we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the class of chordal graphs, split graphs, claw-free graphs and $2K_2$-free graphs.
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Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is $Theta_2^{text{P}}$-complete. They studied the common graph parameters $alpha$ (independence number), $beta$ (vertex cover number), $omega$ (clique number), and $chi$ (chromatic number) for certain variants of the stability problem. We follow their approach and provide a large number of polynomial-time algorithms solving these problems for special graph classes, namely for graphs without edges, complete graphs, paths, trees, forests, bipartite graphs, and co-graphs.
Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). Janos Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an infinite complete graph as a minor, and (ii) a subdivision of the complete graph $K_t$ with multiplicity $t$, for every finite $t$. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite $n$-vertex subgraph has a balanced separator of size $O(sqrt{n})$. The graph is $mathcal{T}_6boxtimes P_{!infty}$, where $mathcal{T}_k$ is the universal treewidth-$k$ countable graph (which we define explicitly), $P_{!infty}$ is the 1-way infinite path, and $boxtimes$ denotes the strong product. More generally, for every positive integer $t$ we construct a countable graph that contains every countable $K_t$-minor-free graph and has the above key properties. Our final contribution is a construction of a countable graph that contains every countable $K_t$-minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably infinite complete graph (implying (ii) above is best possible).
What is the maximum number of copies of a fixed forest $T$ in an $n$-vertex graph in a graph class $mathcal{G}$ as $nto infty$? We answer this question for a variety of sparse graph classes $mathcal{G}$. In particular, we show that the answer is $Theta(n^{alpha_d(T)})$ where $alpha_d(T)$ is the size of the largest stable set in the subforest of $T$ induced by the vertices of degree at most $d$, for some integer $d$ that depends on $mathcal{G}$. For example, when $mathcal{G}$ is the class of $k$-degenerate graphs then $d=k$; when $mathcal{G}$ is the class of graphs containing no $K_{s,t}$-minor ($tgeq s$) then $d=s-1$; and when $mathcal{G}$ is the class of $k$-planar graphs then $d=2$. All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.
Given a family $mathcal{I}$ of independent sets in a graph, a rainbow independent set is an independent set $I$ such that there is an injection $phicolon Ito mathcal{I}$ where for each $vin I$, $v$ is contained in $phi(v)$. Aharoni, Briggs, J. Kim, and M. Kim [Rainbow independent sets in certain classes of graphs. arXiv:1909.13143] determined for various graph classes $mathcal{C}$ whether $mathcal{C}$ satisfies a property that for every $n$, there exists $N=N(mathcal{C},n)$ such that every family of $N$ independent sets of size $n$ in a graph in $mathcal{C}$ contains a rainbow independent set of size $n$. In this paper, we add two dense graph classes satisfying this property, namely, the class of graphs of bounded neighborhood diversity and the class of $r$-powers of graphs in a bounded expansion class.
Let $T$ be a right exact functor from an abelian category $mathscr{B}$ into another abelian category $mathscr{A}$. Then there exists a functor ${bf p}$ from the product category $mathscr{A}timesmathscr{B}$ to the comma category $(Tdownarrowmathscr{A})$. In this paper, we study the property of the extension closure of some classes of objects in $(Tdownarrowmathscr{A})$, the exactness of the functor ${bf p}$ and the detail description of orthogonal classes of a given class ${bf p}(mathcal{X},mathcal{Y})$ in $(Tdownarrowmathscr{A})$. Moreover, we characterize when special precovering classes in abelian categories $mathscr{A}$ and $mathscr{B}$ can induce special precovering classes in $(Tdownarrowmathscr{A})$. As an application, we prove that under suitable cases, the class of Gorenstein projective left $Lambda$-modules over a triangular matrix ring $Lambda=left(begin{smallmatrix}R & M O & S end{smallmatrix} right)$ is special precovering if and only if both the classes of Gorenstein projective left $R$-modules and left $S$-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
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