arXiv subjects

Tala Eagling-Vose

Publications and source records attributed to Tala Eagling-Vose.

10 recordsLinked to original sources

arXiv2026

$O(n +f(k))$: Truly Linear FPT

Parameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter $k$, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size $n$ is unaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. $f(k) \cdot n^c$) or additively (i.e. $f(k) + n^c$). But what if the constant $c$ is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT) -- that is $O(n)+f(k)$ -- and show that it is a strict subset of Linear FPT (LFPT) -- that is $O(n) \cdot f(k)$ -- via diagonalization. Populating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, $(n-k)$-Coloring, Diverse Pair of Matchings, $k$-Path, and $H$-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth- and breadth-first search. For parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.

cs.CC↗

arXiv2026

On Detecting $H$-Induced Minors for Small $H$

We consider the $H$-Induced Minor problem: for a fixed graph~$H$, decide whether a given graph $G$ contains $H$ as an induced minor. While the problem is known to be NP-complete for some trees~$H$ on more than $2^{300}$ vertices, the complexity for small trees remains unresolved. In particular, the case where $H$ is the $7$-vertex tree consisting of a path on five vertices with a pendant vertex attached to the second and fourth vertex was a long-standing open problem. We show that this case is polynomial-time solvable by developing algorithms that detect a sequence of carefully chosen substructures. Complementing this, we prove that detecting some of these substructures individually is NP-hard. We also give polynomial-time algorithms for three cases where $H$ is a graph on five vertices (that is not a tree). In this way, we completed the classification of $H$-Induced Minor for graphs $H$ on five vertices and answered an open problem of Dallard, Dumas, Hilaire and Perez (2025).

math.CO↗

arXiv2026

Optimal b-Colourings and Fall Colourings in $H$-Free Graphs

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in $H$-free graphs. For Tight b-Chromatic Number in $H$-free graphs, we develop a general technique to determine new graphs $H$, for which the problem is polynomial-time solvable, and we also determine new graphs $H$, for which the problem is still NP-complete. We show, for the first time, the existence of a graph $H$ such that in $H$-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

math.CO↗

arXiv2026

Steiner Forest for $H$-Subgraph-Free Graphs

Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and combinatorial results: Algorithms: We identify two new classes of graph-theoretical structures that make it possible to solve Steiner Forest in polynomial time. Roughly speaking, our algorithms handle the following cases: (1) a set $X$ of vertices of bounded size that are pairwise connected by subgraphs of treewidth $2$ or bounded size, possibly together with an independent set of arbitrary size that is connected to $X$ in an arbitrary way; (2) a set $X$ of vertices of arbitrary size that are pairwise connected in a cyclic manner by subgraphs of treewidth $2$ or bounded size. Hardness results: We show that Steiner Forest remains NP-complete for graphs with 2-deletion set number $3$. (The $c$-deletion set number is the size of a smallest cutset $S$ such that every component of $G-S$ has at most $c$ vertices.) Combinatorial results: To establish the dichotomy, we perform a delicate graph-theoretic analysis showing that if $H$ is a path or a subdivided claw, then excluding $H$ as a subgraph either yields one of the two algorithmically favourable structures described above, or yields a graph class for which NP-completeness of Steiner Forest follows from either our new hardness result or a previously known one. Along the way to classifying the hardness for excluded subgraphs, we establish a dichotomy for graphs with $c$-deletion set number at most $k$. Specifically, our results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2$ and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

math.CO↗

arXiv2025

Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains to consider when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided ``H''-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ that is a star, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases $\mathbb{H}_0$ and $\mathbb{H}_1$. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.

math.CO↗

arXiv2025

Finding $d$-Cuts in Probe $H$-Free Graphs

For an integer $d\geq 1$, the $d$-Cut problem is that of deciding whether a graph has an edge cut in which each vertex is adjacent to at most $d$ vertices on the opposite side of the cut. The $1$-Cut problem is the well-known Matching Cut problem. The $d$-Cut problem has been extensively studied for $H$-free graphs. We extend these results to the probe graph model, where we do not know all the edges of the input graph. For a graph $H$, a partitioned probe $H$-free graph $(G,P,N)$ consists of a graph $G=(V,E)$, together with a set $P\subseteq V$ of probes and an independent set $N=V\setminus P$ of non-probes such that we can change $G$ into an $H$-free graph by adding zero or more edges between vertices in $N$. For every graph $H$ and every integer $d\geq 1$, we completely determine the complexity of $d$-Cut on partitioned probe $H$-free graphs.

cs.DS↗

arXiv2025

Bounding Width on Graph Classes of Constant Diameter

We determine if the width of a graph class ${\cal G}$ changes from unbounded to bounded if we consider only those graphs from ${\cal G}$ whose diameter is bounded. As parameters we consider treedepth, pathwidth, treewidth and clique-width, and as graph classes we consider classes defined by forbidding some specific graph $F$ as a minor, induced subgraph or subgraph, respectively. Our main focus is on treedepth for $F$-subgraph-free graphs of diameter at most~$d$ for some fixed integer $d$. We give classifications of boundedness of treedepth for $d\in \{4,5,\ldots\}$ and partial classifications for $d=2$ and $d=3$.

cs.DM↗

arXiv2025

Finding d-Cuts in Claw-free Graphs

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets $B$ and $R$ such that the edges between $B$ and $R$ form a matching, that is, every vertex in $B$ has at most one neighbour in $R$, and vice versa. If for some integer $d\geq 1$, we allow every neighbour in $B$ to have at most $d$ neighbours in $R$, and vice versa, we obtain the more general problem $d$-Cut. It is known that $d$-Cut is NP-complete for every $d\geq 1$. However, for claw-free graphs, it is only known that $d$-Cut is polynomial-time solvable for $d=1$ and NP-complete for $d\geq 3$. We resolve the missing case $d=2$ by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every $d\geq 2$, $d$-Cut, restricted to claw-free graphs of maximum degree $p$, is constant-time solvable if $p\leq 2d+1$ and NP-complete if $p\geq 2d+3$. Moreover, in the former case, we can find a $d$-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to $S_{1^t,l}$-free graphs where $S_{1^t,l}$ is the graph obtained from a star on $t+2$ vertices by subdividing one of its edges exactly $l$ times.

math.CO↗

arXiv2025

Complexity Framework For Forbidden Subgraphs V: Beyond Simple Graphs

We continue the study of the recently-introduced C123-framework, for (simple) graph problems restricted to inputs specified by the forbidding of some finite set of subgraphs, to more general graph problems possibly involving multiedges and self-loops. We study specifically the problems Multigraph Matching Cut, Multigraph d-Cut and Partially Reflexive Stable Cut in this connection. The last may be seen as a Surjective Homomorphism problem to a path P_3 in which both leaves are looped while the interior vertex is loopless. We consider also another family of Surjective Homomorphism problems to a cycle in which only one vertex is loopless. When one forbids a single (simple) subgraph, our first three problems exhibit the same complexity behaviour as C123-problems, but on finite sets of forbidden subgraphs, the classification appears more complex. While Multigraph Matching Cut and Multigraph d-Cut have the same classification as C123-problems, already Partially Reflexive Stable Cut fails to have. This is witnessed by forbidding as subgraphs both C_3 and H_1. Indeed, the difference of behaviour occurs only around pendant subdivisions of nets and pendant subdivisions of H_1. We examine this area in close detail. Our other Surjective Homomorphism problem, ostensibly somewhat similar to Partially Reflexive Stable Cut, behaves very differently when the input is restricted to some class that is H-subgraph-free. For example, it is solvable in polynomial time on any class of bounded degree. Also, its hardness will never be preserved under any form of edge subdivision.

math.CO↗

arXiv2024

Graph Homomorphism, Monotone Classes and Bounded Pathwidth

In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational complexity can be described for all such graph classes, giving a dichotomy between those classes for which the problem is hard and those for which it is easy. In this article, we consider several variants of the homomorphism problem in relation to this framework. It is known that certain homomorphism problems, e.g. $C_5$-Colouring, do not sit in the framework. By contrast, we show that the more general problem of Graph Homomorphism does sit in the framework, with hard cases NP-complete and easy cases in P. We go on to consider several locally constrained variants of the homomorphism problem, namely the locally bijective, surjective and injective variants. Like $C_5$-Colouring, none of these is in the framework. However, where a bounded-degree restrictions are considered, we prove that each of these problems is in our framework, with hard cases NP-complete and easy cases in P Next, we give the first example of a problem in the framework such that hardness is in the polynomial hierarchy above NP. This comes from a list colouring game, realised through first-order logic as quantified constraints. We show that with the additional restriction of bounded alternation, the problem is contained in the framework. The hard cases are $\Pi_{2k}^\mathrm{P}$-complete and the easy cases are in P. Finally, we go on to consider an aforementioned problem from our framework, complete for the second level of the polynomial hierarchy, under the omission in the input of not just a graph, but rather a graph $H$ annotated with the types for each vertex: existential or universal.

cs.CC↗