---
title: 'Self-Deleting Graphs: Traversal-Dependent Complexity'
url: https://www.emergentmind.com/topics/self-deleting-graphs
type: topic
---

# Self-Deleting Graphs: Traversal-Dependent Complexity

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Self-deleting graphs are traversal-dependent graphs in which visiting a vertex triggers irrevocable edge deletions specified by a deletion function. In the formal model studied for pathfinding, a self-deleting graph is an ordered pair $(G,f)$ with $G=(V,E)$ a simple undirected graph and $f:V\to 2^E$ a function such that $f(v)$ is the set of edges deleted after visiting $v$. The resulting feasibility notion is order-sensitive: the graph traversed is not static, because admissible future edges depend on previously visited vertices. This model was introduced by Carmesin, Woller, Parker, Kulich, and Mansouri for Hamiltonian-cycle and travelling-salesperson variants, and was subsequently developed for $s$-$t$ path problems, where it exhibits strong classical hardness, rich parameterized complexity, and a sharp contrast between negative and positive algorithmic results [2507.12047].

## 1. Formal model and traversal-dependent semantics

A self-deleting graph is specified by $(G,f)$, where $G=(V,E)$ and $f:V\to 2^E$. Two standard size measures are
\[
\mu_f := \max_{v\in V}|f(v)|,
\qquad
|f| := \sum_{v\in V}|f(v)|.
\]
The parameter $\mu_f$, usually written $\mu$, is the maximum number of edges deleted by a single visited vertex; $|f|$ is the total number of encoded deletions [2507.12047].

For a walk
\[
P=(v_1,e_1,v_2,e_2,\dots,e_{k-1},v_k)
\]
in $G$, with $e_i=\{v_i,v_{i+1}\}\in E$, the walk is called $f$-conforming if
\[
\forall\, 1\le j\le i\le k-1:\ e_i\notin f(v_j).
\]
Equivalently, once a vertex $v_j$ has been visited, every edge in $f(v_j)$ is permanently unavailable for the remainder of the traversal. A key feature is that deleted edges need not be incident to the triggering vertex. That non-locality is the main source of combinatorial difficulty: a local routing choice can invalidate edges arbitrarily far away in the graph [2507.12047].

The model becomes substantially simpler under a strong locality restriction. If every edge in $f(v)$ is incident to $v$, then self-deleting $s$-$t$ path reduces to ordinary directed path finding: orient an undirected edge $\{u,v\}$ as $(u,v)$ if it is deleted at $v$ but not at $u$, and remove the edge if both endpoints delete it. In that special case, path finding is linear-time. This isolates the essential hardness mechanism: allowing deletion of non-incident edges [2507.12047].

## 2. Path problems on self-deleting graphs

The central decision problem is **Self-Deleting $s$-$t$-Path**. Its input is a self-deleting graph $(G,f)$ and two distinct vertices $s,t\in V$. The question is whether there exists a simple $f$-conforming path
\[
P=(v_1,e_1,v_2,\dots,e_{k-1},v_k)
\]
with $v_1=s$, $v_k=t$, and
\[
\forall\,1\le j\le i\le k-1:\ e_i\notin f(v_j).
\]
The condition states that no traversed edge has already been deleted by any previously visited vertex [2507.12047].

The length-bounded variant is **Shortest Self-Deleting $s$-$t$-Path**. Its input augments $(G,f),s,t$ with a positive integer $k$, and asks whether there exists an $f$-conforming simple $s$-$t$ path using at most $k$ vertices, equivalently at most $k-1$ edges. In the parameterized analysis of the problem, the parameter $k$ refers exactly to this vertex bound [2507.12047].

The reduction framework for the basic NP-hardness result already shows the semantic interaction between traversal and deletion. Variable gadgets are 4-cycles whose two branches encode truth assignments, and clause gadgets are $2\times |C|$ ladders whose vertical edges represent literals. Choosing the true or false branch in a variable gadget deletes literal-edges for the opposite polarity in later clause gadgets. A satisfying assignment corresponds to an $f$-conforming $s$-$t$ path, and any $f$-conforming path induces a satisfying assignment. The same construction can be inflated so that every vertex deletes at most one edge, preserving hardness under an extremely sparse deletion pattern [2507.12047].

## 3. Hardness and parameterized complexity

The basic decision problem is NP-hard even under unusually strong restrictions. In particular, Self-Deleting $s$-$t$-Path remains NP-hard when the underlying graph is simultaneously outerplanar, bipartite, of maximum degree $3$, of bandwidth $2$, and satisfies $|f(v)|\le 1$ for every vertex. The reduction is from 3SAT, and the constructed graph is a subgraph of a ladder, showing that traversal dependence alone suffices to encode NP-hardness on very simple graph classes [2507.12047].

The length-bounded problem has a sharp parameterized boundary. **Shortest Self-Deleting $s$-$t$-Path**, parameterized by the number $k$ of vertices in the sought path, is W[1]-complete. W[1]-hardness is obtained from **Multicolored Clique** by forcing the path to choose exactly one short subpath per color class, with deletion sets eliminating incompatible choices. Membership in W[1] is shown by a reverse reduction to Multicolored Clique via a layered compatibility graph [2507.12047].

The unbounded decision problem is also hard for several standard structural parameters. Self-Deleting $s$-$t$-Path is W[1]-complete when parameterized by vertex cover number, feedback vertex set number, and distance to linear forest; it is also W[1]-complete when parameterized by treedepth or by vertex integrity. The treedepth and vertex-integrity results exploit the fact that those parameters bound the maximum length of any simple path: if $\operatorname{td}(G)=\alpha$, then no path has more than $2^\alpha$ vertices, and if $\operatorname{vi}(G)=\beta$, then any path has at most $\beta^2+2$ vertices. These bounds yield algorithms of time $n^{O(2^\alpha)}$ and $n^{O(\beta^2)}$, and under ETH the exponents are essentially optimal [2507.12047].

A common misconception is that sparse or tree-like structure should trivialize the problem. The results do not support that view. Even distance to linear forest is insufficient for fixed-parameter tractability, and hardness survives under bandwidth $2$ and outerplanarity. At the same time, the difficulty is not uniform across all variants: on cactus graphs, existence of an $f$-conforming $s$-$t$ path is solvable in linear time, whereas **Shortest** Self-Deleting $s$-$t$-Path remains NP-hard on cactus graphs even with $\mu\le 1$ [2507.12047].

## 4. Fixed-parameter tractability and algorithmic techniques

The parameter $\mu=\max_v |f(v)|$ does not by itself yield tractability. NP-hardness persists even when $\mu\le 1$, so neither the decision nor the shortest-path variant becomes fixed-parameter tractable under parameterization by $\mu$ alone [2507.12047].

The main positive result is an FPT algorithm for the combined parameter $(k,\mu)$. For **Shortest Self-Deleting $s$-$t$-Path**, there is a randomized algorithm with one-sided error running in time
\[
2^{O(k\log \mu)}\cdot (n+m)\cdot \log(1/\varepsilon),
\]
and deterministic algorithms running in time
\[
2^{O(\mu k)}(n+m)\log n
\quad\text{or}\quad
2^{O(k\log(k\mu))}(n+m)\log n,
\]
depending on the derandomization method. The method is a color-coding scheme tailored to deletions: for a candidate path $P$, only the path edges and the edges in the deletion sets of visited vertices matter, and that relevant set has size $O(k\mu)$. A coloring $\chi:E(G)\to [q]$ is sought such that the path is half-rainbow; then dynamic programming over states $(v,Q')$, where $Q'$ is a set of still-available colors, tests whether a $\chi$-compliant path exists [2507.12047].

The same $(k,\mu)$ algorithm becomes a general engine once one can bound the length of a shortest conforming path as a function of structural parameters. The key observation is that if $P$ is a shortest $f$-conforming $s$-$t$ path on $k$ vertices and $G_P=G[V(P)]$, then
\[
|E(G_P)| \le k\mu + (k-1) \le k(\mu+1),
\]
because every chord in $G_P$ must be deleted by some vertex of $P$. Combined with density lower bounds for traceable graphs, this yields FPT results for parameter pairs $(\alpha,\mu)$ when $\alpha$ is any of cluster vertex deletion number, neighborhood diversity, distance to cograph, modular-width, maximum induced matching size, or shrub-depth [2507.12047].

There are also more classical positive results. Parameterization by feedback edge number gives an algorithm running in time
\[
O\!\left(2^{\operatorname{fen}(G)}(n+m+|f|)\right),
\]
because the number of simple $s$-$t$ paths is at most $2^{\operatorname{fen}(G)}$. Combining explicit path-length bounds with the $(k,\mu)$ algorithm gives running times
\[
2^{O(\mu\cdot \operatorname{vc}(G))}(n+m)\log n,\qquad
2^{O(\mu\cdot \operatorname{vi}(G)^2)}(n+m)\log n,\qquad
2^{O(\mu\cdot 2^{\operatorname{td}(G)})}(n+m)\log n
\]
for parameterizations by vertex cover, vertex integrity, and treedepth together with $\mu$ [2507.12047].

## 5. Restricted classes, preprocessing, and lower bounds

Restricted graph classes expose a pronounced separation between feasibility and optimization. On cactus graphs, the existence problem is linear-time solvable via a reduction to 2SAT on the block-cut structure: each cycle contributes a binary choice between its two arcs, and deletion interactions become implications. By contrast, the shortest-path variant is NP-hard on the same class, which shows that minimizing length reintroduces global combinatorial coupling even where mere existence is tractable [2507.12047].

Kernelization results are mostly negative. Unless $\textsf{NP}\subseteq \textsf{coNP}/\text{poly}$, Self-Deleting $s$-$t$-Path does not admit a polynomial kernel when parameterized by vertex cover number together with $\mu$, even on 2-outerplanar graphs; it also has no polynomial kernel for vertex cover alone, even with $\mu=1$, again on 2-outerplanar graphs; and it has no polynomial kernel for $\mu$ alone on cliques. These lower bounds are obtained by OR-cross-composition from 3SAT [2507.12047].

The negative picture is not absolute. Positive kernelization results include a linear kernel with $O(\operatorname{fen}(G))$ vertices and edges for parameterization by feedback edge number, a linear kernel with $O(\operatorname{vc}(G))$ vertices on outerplanar graphs, and a Turing kernel with $O(\mu)$ vertices on cliques. The clique result exploits a strong structural restriction: for each possible second vertex $v$ on a shortest conforming path, it suffices to inspect a subinstance on
\[
X^v=\{s,t,v\}\cup \{u:\{s,u\}\in f(s)\},
\]
which has size at most $\mu+3$ [2507.12047].

These results delimit a characteristic algorithmic profile of self-deleting graphs. Sparse or low-width structure alone is generally insufficient; useful positive results arise either from strong path-length bounds, from combining structure with bounded deletion arity $\mu$, or from graph classes whose simple-path space can be explicitly controlled [2507.12047].

## 6. Related models and terminological scope

Within algorithmic graph theory, self-deleting graphs belong to a broader family of dynamic and traversal-dependent models. The closest comparisons are temporal graphs and time-dependent networks, Optional Choice Reachability, path problems with forbidden pairs, paths with forbidden edge transitions, and arc-dependent networks. The common theme is that reachability depends on history or state, but self-deleting graphs are distinctive in expressing that dependence as global edge deletion triggered by vertex visits rather than by time labels or local transition constraints [2507.12047].

The phrase *self-deleting graph* is also used in distinct senses outside traversal-dependent pathfinding. In spectral graph theory, 1-walk-regular graphs can be “self-deleting” in the sense that deleting an edge, or deleting the edge set of a graph embedded in a clique, yields families of pairwise non-isomorphic graphs that are cospectral for adjacency, Laplacian, unsigned Laplacian, and normalized Laplacian matrices [2302.03854]. In percolation theory, a related deletion-driven notion studies the graph obtained after removing all vertices in infinite percolation clusters, with results on the critical threshold of the remaining graph and on self-destructive percolation on transitive unimodular non-amenable graphs [1302.6870]. In graph generation, GrIDDD reformulates discrete graph diffusion so that the graph can monotonically grow or shrink through node insertion and deletion during the diffusion process, using DEL and DEL$^*$ states and node activation times [2506.15725].

A plausible implication is that “self-deleting graph” is best treated as a family resemblance term rather than a single canonical concept. In the formal pathfinding literature, however, it denotes the specific traversal-dependent model $(G,f)$ with $f$-conforming paths, and it is in that sense that the current complexity-theoretic landscape is most fully developed [2507.12047].

Source: https://www.emergentmind.com/topics/self-deleting-graphs