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We provide theoretical insights around the cutwidth of a graph and the One-Sided Crossing Minimization (OSCM) problem. OSCM was posed in the Parameterized Algorithms and Computational Experiments Challenge 2024, where the cutwidth of the input graph was the parameter in the parameterized track. We prove an asymptotically sharp upper bound on the size of a graph in terms of its order and cutwidth. As the number of so-called unsuited pairs is one of the factors that determine the difficulty of an OSCM instance, we provide a sharp upper bound on them in terms of the order $n$ and the cutwidth of the input graph. If the cutwidth is bounded by a constant, this implies an $\mathcal{O}(2^n)$-time algorithm, while the trivial algorithm has a running time of $\mathcal{O}(2^{n^2})$. At last, we prove structural properties of the so-called crossing numbers in an OSCM instance.

For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $\lambda$, minimum degree at least $\delta$, and maximum degree at most $\Delta$, where $0\leq\delta \frac{dn}{\sqrt{2}}. \end{cases}$$ Our results are based on a smoothing technique relating the degree deviation and the largest eigenvalue to low-dimensional non-linear optimization problems.

Confirming a conjecture posed by Caro, it was shown by Chen and Yu that every graph $G$ with $n$ vertices and at most $2n-4$ edges has a stable cutset, which is a stable set of vertices whose removal disconnects the graph. Le and Pfender showed that all graphs with $n$ vertices and $2n-3$ edges without stable cutset arise recursively glueing together triangles and triangular prisms along an edge or triangle. Le and Pfender's proof contains a gap, which we fill in the present article.

We propose the conjecture that every graph $G$ of order $n$ with less than $3n-6$ edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible. We verify the conjecture for planar graphs and show that every graph $G$ of order $n$ with less than $\frac{11}{5}n-\frac{18}{5}$ edges has a vertex cut that induces a forest.

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a conjecture of Nikiforov, we show $\lambda-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$. For our result, we show that the largest eigenvalue of a graph that arises from a bipartite graph with $m_{A,B}$ edges by adding $m_A$ edges within one of the two partite sets is at most $\sqrt{m_A+m_{A,B}+\sqrt{m_A^2+2m_Am_{A,B}}}$, which is a common generalization of results due to Stanley and Bhattacharya, Friedland, and Peled.

A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.

Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of forests, we study maximum common induced subforests and minimum common induced superforests. We show that finding a maximum subforest is NP-hard already for two subdivided stars while finding a minimum superforest is tractable for two trees but NP-hard for three trees. For a given set of $k$ trees, we present an efficient greedy $\left(\frac{k}{2}-\frac{1}{2}+\frac{1}{k}\right)$-approximation algorithm for the minimum superforest problem. Finally, we present a polynomial time approximation scheme for the maximum subforest problem for any given set of forests.

A common subgraph of two graphs $G_1$ and $G_2$ is a graph that is isomorphic to subgraphs of $G_1$ and $G_2$. In the largest common subgraph problem the task is to determine a common subgraph for two given graphs $G_1$ and $G_2$ that is of maximum possible size ${\rm lcs}(G_1,G_2)$. This natural problem generalizes the well-studied graph isomorphism problem, has many applications, and remains NP-hard even restricted to unions of paths. We present a simple $4$-approximation algorithm for forests, and, for every fixed $\epsilon\in (0,1)$, we show that, for two given forests $F_1$ and $F_2$ of order at most $n$, one can determine in polynomial time a common subgraph $F$ of $F_1$ and $F_2$ with at least ${\rm lcs}(F_1,F_2)-\epsilon n$ edges. Restricted to instances with ${\rm lcs}(F_1,F_2)\geq cn$ for some fixed positive $c$, this yields a polynomial time approximation scheme. Our approach relies on the approximation of the given forests by structurally simpler forests that are composed of copies of only $O(\log (n))$ different starlike rooted trees and iterative quantizations of the options for the solutions.

One way to define the Matching Cut problem is: Given a graph $G$, is there an edge-cut $M$ of $G$ such that $M$ is an independent set in the line graph of $G$? We propose the more general Conflict-Free Cut problem: Together with the graph $G$, we are given a so-called conflict graph $\hat{G}$ on the edges of $G$, and we ask for an edge-cutset $M$ of $G$ that is independent in $\hat{G}$. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is $\textsf{NP}$-complete even when the maximum degree of $G$ is 5 and $\hat{G}$ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of $G$ as a parameter, and we show $\textsf{W[1]}$-hardness even when $G$ has a feedback vertex set of size one, and the clique cover number of $\hat{G}$ is the parameter. Since the clique cover number of $\hat{G}$ is an upper bound on the independence number of $\hat{G}$ and thus the solution size, this implies $\textsf{W[1]}$-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT., Comment: 13 pages, 3 figures

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases., Comment: 20 pages with references and appendix

Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. For a graph $G$ of order $n$ and maximum degree at most $\Delta$, we show $Mo(G)\leq \frac{\Delta}{2}n^2-(1-o(1))c_{\Delta}n\log(\log(n)),$ where $c_{\Delta}>0$ only depends on $\Delta$ and the $o(1)$ term only depends on $n$. Furthermore, for integers $n_0$ and $\Delta$ at least $3$, we show the existence of a $\Delta$-regular graph of order $n$ at least $n_0$ with $Mo(G)\geq \frac{\Delta}{2}n^2-c'_{\Delta}n\log(n),$ where $c'_{\Delta}>0$ only depends on $\Delta$.

We show that every graph $G$ of maximum degree $\Delta$ and sufficiently large order has a vertex cutset $S$ of order at most $\Delta$ that induces a subgraph $G[S]$ of maximum degree at most $\Delta-3$. For $\Delta\in \{ 4,5\}$, we refine this result by considering also the average degree of $G[S]$. If $G$ has no $K_{r,r}$ subgraph, then we show the existence of a vertex cutset that induces a subgraph of maximum degree at most $\left(1-\frac{1}{{r\choose 2}}\right)\Delta+O(1)$.

Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\Delta$, and $d=\left\lfloor \frac{\Delta m}{\Delta n-m}\right\rfloor$, we show $$irr(G)\leq d(d+1)n+\frac{1}{\Delta}\left(\Delta^2-(2d+1)\Delta-d^2-d\right)m.$$

Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular, for $d_+\in \left(\sqrt{dn},n-1\right]$, we show the existence of a vertex in $G$ of degree between $d_+-\left(\frac{(d_+-d)n}{n-d_++\sqrt{d_+^2-dn}}\right)$ and $d_+$.

Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. They conjectured that $Mo(G)\leq 0.\overline{148}n^3$ for every graph $G$ of order $n$. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter $Mo^\star(G)=\sum\limits_{uv\in E(G)}\big(n-\min\{ d_G(u),d_G(v)\}\big)$. For a graph $G$ of order $n$, they show that $Mo^\star(G)\leq \frac{5}{24}(1+o(1))n^3$. We improve this bound to $Mo^\star(G)\leq \left(\frac{2}{\sqrt{3}}-1\right)n^3$, which is best possible up to terms of lower order. Furthermore, we show that $Mo^\star(G)\leq \left(2\left(\frac{\Delta}{n}\right)^2+\left(\frac{\Delta}{n}\right)-2\left(\frac{\Delta}{n}\right)\sqrt{\left(\frac{\Delta}{n}\right)^2+\left(\frac{\Delta}{n}\right)}\right)n^3$ provided that $G$ has maximum degree $\Delta$.

Do\v{s}li\'{c} et al.~defined the Mostar index of a graph $G$ as $\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Do\v{s}li\'{c} et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}n^3$.

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the best possible results that every graph of minimum degree at least $2$ has a majority $4$-edge-coloring, and that every graph of minimum degree at least $4$ has a majority $3$-edge-coloring. Furthermore, we discuss a natural variation of majority edge-colorings and some related open problems.

The independence number $\alpha(G)$ and the dissociation number ${\rm diss}(G)$ of a graph $G$ are the largest orders of induced subgraphs of $G$ of maximum degree at most $0$ and at most $1$, respectively. We consider possible improvements of the obvious inequality $2\alpha(G)\geq {\rm diss}(G)$. For connected cubic graphs $G$ distinct from $K_4$, we show $5\alpha(G)\geq 3{\rm diss}(G)$, and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle-free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.

The annihilation number $a(G)$ of a graph $G$ is an efficiently computable upper bound on the independence number $\alpha(G)$ of $G$. Recently, Hiller observed that a characterization of the graphs $G$ with $\alpha(G)=a(G)$ due to Larson and Pepper is false. Since the known efficient algorithm recognizing these graphs was based on this characterization, the complexity of recognizing graphs $G$ with $\alpha(G)=a(G)$ was once again open. We show that these graphs can indeed be recognized efficiently. More generally, we show that recognizing graphs $G$ with $\alpha(G)\geq a(G)-\ell$ is fixed parameter tractable using $\ell$ as parameter.

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph $G$ with $n$ vertices, $m$ edges, $k$ components, and $c_1$ induced cycles of length $1$ modulo $3$, we show ${\rm diss}(G)\geq n-\frac{1}{3}\Big(m+k+c_1\Big)$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex-disjoint.

A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most $1$. Computing the dissociation number ${\rm diss}(G)$ of a given graph $G$, defined as the order of a maximum dissociation set in $G$, is algorithmically hard even when $G$ is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple $\frac{4}{3}$-approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality ${\rm diss}(G)\leq\frac{4}{3}\alpha(G-M)$ implicit in their work, where $G$ is a bipartite graph, $M$ is a maximum matching in $G$, and $\alpha(G-M)$ denotes the independence number of $G-M$. We show that the pairs $(G,M)$ for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph $G$ satisfies $\max\{ \alpha(G),2\nu_s(G)\} \leq {\rm diss}(G)\leq \alpha(G)+\nu_s(G)\leq 2\alpha(G)$, where $\nu_s(G)$ denotes the induced matching number of $G$. We show that deciding whether ${\rm diss}(G)$ equals any of the four terms lower and upper bounding ${\rm diss}(G)$ is NP-hard.

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that $d \leq (2n-3)/5$ for every connected $C_4$-free graph of order $n$, diameter $d$, and edge-connectivity at least $3$, which is best possible up to a small additive constant. For edge-connectivity at least $4$, we improve this to $d \leq (n-3)/3$. Furthermore, adapting a construction due to Erd\H{o}s, Pach, Pollack, and Tuza, for an odd prime power $q$ at least $7$, and every positive integer $k$, we show the existence of a connected $C_4$-free graph of order $n=(q^2+q-1)k+1$, diameter $d=4k$, and edge-connectivity $\lambda$ at least $q-6$, in particular, $d\geq 4(n-1)/(\lambda^2+O(\lambda))$.

A classical result of Erd\H{o}s and Gallai determines the maximum size $m(n,\nu)$ of a graph $G$ of order $n$ and matching number $\nu n$. We show that $G$ has factorially many maximum matchings provided that its size is sufficiently close to $m(n,\nu)$.

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(c_{d,\Delta}-O\left(\frac{1}{n}\right)\right)m(G)$, where $c_{d,\Delta}=d+\Omega\left(\frac{1}{\Delta}\right)$ for fixed $d$, that is, this number visibly deviates from its expected value when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $\Delta\leq 2$, we present more detailed results.

For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every $k$-edge colored color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$. For a perfect matching $M$ of $K_{2kn}$, a natural measure for the total deviation of $M$ from being color-balanced is $f(M)=\sum\limits_{i=1}^k|m_i(M)-n|$. While not every color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$, that is, a perfect matching with $f(M)=0$, we prove the existence of a perfect matching $M$ with $f(M)=O\left(k\sqrt{kn\ln(k)}\right)$ for general $k$ and $f(M)\leq 2$ for $k=3$; the case $k=2$ has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.

We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing XP algorithms. The latter of these results answers a question of Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016]. Further, given an integer $k$, a graph $G=(V,E)$, and $r \in V$, we consider the problem of finding two arc-disjoint $(r,k)$-safe spanning trees. We show that this problem is also FPT with parameter $k$, again improving on a previous XP algorithm. Our main technical contribution is to prove that the existence of such spanning substructures is equivalent to the existence of substructures with size and maximum (out-)degree both bounded by a (linear or quadratic) function of $k$, which may be of independent interest., Comment: 20 pages, 1 figure

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavi\v{c} and \v{S}parl (arXiv:2011.01635v1) define the distance-unbalancedness ${\rm uB}(G)$ of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of distinct vertices $u$ and $v$ of $G$. For positive integers $n$ up to $15$, they determine the trees $T$ of fixed order $n$ with the smallest and the largest values of ${\rm uB}(T)$, respectively. While the smallest value is achieved by the star $K_{1,n-1}$ for these $n$, which we then proved for general $n$ (Minimum distance-unbalancedness of trees, Journal of Mathematical Chemistry, DOI 10.1007/s10910-021-01228-4), the structure of the trees maximizing the distance-unbalancedness remained unclear. For $n$ up to $15$ at least, all these trees were subdivided stars. Contributing to problems posed by Miklavi\v{c} and \v{S}parl, we show $$\max\Big\{{\rm uB}(T):T\mbox{ is a tree of order }n\Big\} =\frac{n^3}{2}+o(n^3)$$ and $$\max\Big\{{\rm uB}(S(n_1,\ldots,n_k)):1+n_1+\cdots+n_k=n\Big\} =\left(\frac{1}{2}-\frac{5}{6k}+\frac{1}{3k^2}\right)n^3+O(kn^2),$$ where $S(n_1,\ldots,n_k)$ is the subdivided star such that removing its center vertex leaves paths of orders $n_1,\ldots,n_k$.

For a fixed positive $\epsilon$, we show the existence of a constant $C_\epsilon$ with the following property: Given a $\pm1$-edge-labeling $c:E(K_n)\to \{ -1,1\}$ of the complete graph $K_n$ with $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$ of maximum degree $\Delta$, one can determine in polynomial time an isomorphic copy $F'$ of $F$ in $K_n$ with $|c(E(F'))|\leq \left(\frac{3}{4}+\epsilon\right)\Delta+C_\epsilon.$ Our approach is based on the method of conditional expectation.

For a complete graph $K_n$ of order $n$, an edge-labeling $c:E(K_n)\to \{ -1,1\}$ satisfying $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$, we consider the problem to minimize $|c(E(F'))|$ over all isomorphic copies $F'$ of $F$ in $K_n$. In particular, we ask under which additional conditions there is a zero-sum copy, that is, a copy $F'$ of $F$ with $c(E(F'))=0$. We show that there is always a copy $F'$ of $F$ with $|c(E(F'))|\leq \Delta(F)+1$, where $\Delta(F)$ is the maximum degree of $F$. We conjecture that this bound can be improved to $|c(E(F'))|\leq (\Delta(F)-1)/2$ and verify this for $F$ being the star $K_{1,n-1}$. Under some simple necessary divisibility conditions, we show the existence of a zero-sum $P_3$-factor, and, for sufficiently large $n$, also of a zero-sum $P_4$-factor.

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavi\v{c} and \v{S}parl (arXiv:2011.01635v1) define the distance-unbalancedness of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of distinct vertices $u$ and $v$ of $G$. Confirming one of their conjectures, we show that the stars minimize the distance-unbalancedness among all trees of a fixed order.

Answering a question posed by Caro, Hansberg, Lauri, and Zarb, we show that for every positive integer $n$ and every function $\sigma\colon E(K_{4n})\to\{-1,1\}$ with $\sigma\left(E(K_{4n})\right)=0$, there is a perfect matching $M$ in $K_{4n}$ with $\sigma(M)=0$. Strengthening a result of Caro and Yuster, we show that for every positive integer $n$ and every function $\sigma\colon E(K_{4n})\to\{-1,1\}$ with $\left|\sigma\left(E(K_{4n})\right)\right|

A set $S$ of vertices of a graph $G$ is exponentially independent if, for every vertex $u$ in $S$, $$\sum\limits_{v\in S\setminus \{ u\}}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}<1,$$ where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u$ and $v$ in the graph $G-(S\setminus \{ u,v\})$. The exponential independence number $\alpha_e(G)$ of $G$ is the maximum order of an exponentially independent set in $G$. In the present paper we present several bounds on this parameter and highlight some of the many related open problems. In particular, we prove that subcubic graphs of order $n$ have exponentially independent sets of order $\Omega(n/\log^2(n))$, that the infinite cubic tree has no exponentially independent set of positive density, and that subcubic trees of order $n$ have exponentially independent sets of order $(n+3)/4$.

We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq \tau(u)\}$ for every positive integer $t$, and $\tau:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $\tau$ satisfying $\tau(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.

For a graph $G$, two dominating sets $D$ and $D'$ in $G$, and a non-negative integer $k$, the set $D$ is said to $k$-transform to $D'$ if there is a sequence $D_0,\ldots,D_\ell$ of dominating sets in $G$ such that $D=D_0$, $D'=D_\ell$, $|D_i|\leq k$ for every $i\in \{ 0,1,\ldots,\ell\}$, and $D_i$ arises from $D_{i-1}$ by adding or removing one vertex for every $i\in \{ 1,\ldots,\ell\}$. We prove that there is some positive constant $c$ and there are toroidal graphs $G$ of arbitrarily large order $n$, and two minimum dominating sets $D$ and $D'$ in $G$ such that $D$ $k$-transforms to $D'$ only if $k\geq \max\{ |D|,|D'|\}+c\sqrt{n}$. Conversely, for every hereditary class ${\cal G}$ that has balanced separators of order $n\mapsto n^\alpha$ for some $\alpha<1$, we prove that there is some positive constant $C$ such that, if $G$ is a graph in ${\cal G}$ of order $n$, and $D$ and $D'$ are two dominating sets in $G$, then $D$ $k$-transforms to $D'$ for $k=\max\{ |D|,|D'|\}+\lfloor Cn^\alpha\rfloor$.

A bihole in a bipartite graph $G$ with partite sets $A$ and $B$ is an independent set $I$ in $G$ with $|I\cap A|=|I\cap B|$. We prove lower bounds on the largest order of biholes in balanced bipartite graphs subject to conditions involving the vertex degrees and the average degree.

The tree breadth ${\rm tb}(G)$ of a connected graph $G$ is the smallest non-negative integer $\rho$ such that $G$ has a tree decomposition whose bags all have radius at most $\rho$. We show that, given a connected graph $G$ of order $n$ and size $m$, one can construct in time $O(m\log n)$ an additive tree $O\big({\rm tb}(G)\log n\big)$-spanner of $G$, that is, a spanning subtree $T$ of $G$ in which $d_T(u,v)\leq d_G(u,v)+O\big({\rm tb}(G)\log n\big)$ for every two vertices $u$ and $v$ of $G$. This improves earlier results of Dragan and K\"{o}hler (Algorithmica 69 (2014) 884-905), who obtained a multiplicative error of the same order, and of Dragan and Abu-Ata (Theoretical Computer Science 547 (2014) 1-17), who achieved the same additive error with a collection of $O(\log n)$ trees.

A matching $M$ in a graph $G$ is acyclic if the subgraph of $G$ induced by the set of vertices that are incident to an edge in $M$ is a forest. We prove that every graph with $n$ vertices, maximum degree at most $\Delta$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{\Delta^2},$ and we explain how to find such an acyclic matching in polynomial time.

Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $\delta(G) \alpha(G)\leq \Delta(G) \mu(G)$ for every graph $G$ with minimum degree $\delta(G)$, maximum degree $\Delta(G)$, independence number $\alpha(G)$, and matching number $\mu(G)$. Answering some problems they posed, we characterize the extremal graphs for $\delta(G)<\Delta(G)$ as well as for $\delta(G)=\Delta(G)=3$.

We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph. We prove that $\gamma(G)\leq \left(1+\frac{2(\Delta-1)}{\Delta 2^{\Delta}}\right)\gamma_e(G)$ for general $\Delta\geq 1$, and $\gamma(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)\gamma_e(G)$ for $\Delta=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.

The edge domination number $\gamma_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on $\gamma_e(G)$ for regular and non-regular graphs $G$ in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (Approximating weighted induced matchings, Discrete Applied Mathematics 243 (2018) 304-310) provide an approximation algorithm with ratio $\Delta$ for the weighted version of the induced matching problem on graphs of maximum degree $\Delta$. Their approach is based on an integer linear programming formulation whose integrality gap is at least $\Delta-1$, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most $\frac{5}{8}\Delta+O(1)$, and that also the approximation ratio can be improved at least to this value. We provide primal-dual approximation algorithms with ratios $(1-\epsilon) \Delta + \frac{1}{2}$ for general $\Delta$ with $\epsilon \approx 0.02005$, and $\frac{7}{3}$ for $\Delta=3$. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of size at least $(n-1) / 3$ except for two exceptional cubic graphs of order $14$ and $20$.

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $\alpha_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. It is known that $\alpha(G)\leq \alpha_b(G)\leq 4\alpha(G)$ for every connected graph $G$, where $\alpha(G)$ is the independence number of $G$. If $G$ has girth $g$ and minimum degree $\delta$, we show that $\alpha_b(G)\leq 2\alpha(G)$ provided that $g\geq 6$ and $\delta\geq 3$ or that $g\geq 4$ and $\delta\geq 5$. Furthermore, we show that, for every positive integer $k$, there is a connected graph $G$ of girth at least $k$ and minimum degree at least $k$ such that $\alpha_b(G)\geq 2\left(1-\frac{1}{k}\right)\alpha(G)$. Our results imply that lower bounds on the girth and the minimum degree of a connected graph $G$ can lower the fraction $\frac{\alpha_b(G)}{\alpha(G)}$ from $4$ below $2$, but not any further.