In this paper we introduce "graph linear notation" -- a complete graph invariant -- which is positioned as an alternative definition for the finite graphs. This invariant is constructed using an algorithm similar to the algorithm of finding canonical forms of graphs. Storing graph linear notation instead of a regular graph allows us to greatly simplify two major problems: the construction of illustrations for graphs with regards to possible graph symmetries, and the comparison of two graphs for isomorphism. We also demonstrate the transferability to the graph linear notations such classical graph theory concepts as colourings and graph paths.

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a Lawrence oriented matroid, thus confirming the conjecture in this case.

The saturation number $μ^*(G)$ of a graph $G$ is the minimum cardinality of a maximal matching, and $H(G)$ is its harmonic index. TxGraffiti conjectured in 2023 that $μ^*(G) \le H(G)$ for every nontrivial connected graph $G$, and Bıyıkoğlu refuted this by showing that the ratio $μ^*(G)/H(G)$ can be made arbitrarily large. Restricting to trees bounds the ratio sharply. Every nontrivial tree $T$ satisfies $μ^*(T) < \frac{3}{2} H(T)$, with the constant $3/2$ best possible. A complementary bound $H(G) < 4μ^*(G)$ holds for every graph with an edge, so on a nontrivial tree the saturation number is pinned to $\frac{1}{4} H(T) < μ^*(T) < \frac{3}{2} H(T)$, both constants best possible. The friendship graph $F_4$ is a smallest counterexample to the conjecture, on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices. For each positive integer $m$ a family of graphs with $m$ hubs has ratio approaching $m+1$, while the conjecture holds whenever all vertices have equal degree. Both invariants arise in applications, the harmonic index as a molecular descriptor and the saturation number as a measure of adsorption inefficiency, and the bounds estimate the latter, which is NP-hard to compute, by the former, which is computable in linear time.

We study a class of bilevel interdiction problems in which the follower's optimization problem is computationally intractable. Motivated by network defense applications, we introduce the Randomized Max-Vertex-Coverage Interdiction (RMVCI) problem under matroid constraints. In this zero-sum Stackelberg game, the leader commits to a randomized interdiction strategy over feasible vertex subsets, while the follower, after observing the induced protection probabilities, chooses a matroid-constrained attack to maximize the expected coverage of network edges. The main challenge stems from the fact that the follower's problem is a matroid-constrained maximum vertex coverage problem and is therefore NP-hard. To address this difficulty, we first develop a general approximation framework for bilevel optimization problems with hard follower responses. The framework is based on replacing the follower's value function by a surrogate objective that approximates the follower's optimal payoff while preserving tractability of the leader's optimization problem. For the RMVCI problem, we formulate the follower's problem as an integer linear program, establish a tight integrality gap of $4/3$ for its linear relaxation, and derive a polynomial-time $4/3$-approximation algorithm via pipage rounding. We then show that a carefully designed surrogate objective admits a marginal-probability reformulation that transforms the randomized interdiction problem into a tractable optimization problem over the leader's matroid polytope. This yields a polynomial-time $2$-approximation algorithm for RMVCI under general matroid constraints. Beyond the specific application studied here, our results provide a new perspective on approximation methods for {general} bilevel optimization problems.

We prove that the poset of $q$-decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary $q>0$, and for any positive rational number $q$, we determine the number of coverings, intervals and meet-irreducible elements. The latter present the same structure as words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.

One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is ${O(|V(H)|^9\operatorname{poly~log} |V(H)|)}$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(|V(H)|\log^2 |V(H)|)$, which is a $\log|V(H)|$ factor away being optimal.

Any continuous piecewise-linear function $F\colon \mathbb{R}^{n}\to \mathbb{R}$ can be represented as a linear combination of $\max$ functions of at most $n+1$ affine-linear functions. In our previous paper [``Representing piecewise linear functions by functions with small arity'', AAECC, 2023], we showed that this upper bound of $n+1$ arguments is tight. In the present paper, we extend this result by establishing a correspondence between the function $F$ and the minimal number of arguments that are needed in any such decomposition. We show that the tessellation of the input space $\mathbb{R}^{n}$ induced by the function $F$ has a direct connection to the number of arguments in the $\max$ functions.