Liar's Vertex-Edge Domination

• Liar’s vertex-edge dominating set is defined on a graph such that every edge is locally monitored by at least two vertices and every pair of edges by at least three, enhancing fault tolerance.
• The concept extends traditional vertex-edge domination by combining redundancy with pairwise disambiguation to counteract false reports in network monitoring.
• Algorithmic approaches range from linear-time exact solutions on trees and proper interval graphs to logarithmic-factor approximations on general graphs, reflecting diverse complexity boundaries.

Liar’s vertex-edge dominating set is a fault-tolerant edge-monitoring concept on a graph $G=(V,E)$. For an edge $e=xy$, its closed neighbourhood is $N_G[e]=N_G[x]\cup N_G[y]$, and a set $L\subseteq V$ is a liar’s vertex-edge dominating set if every edge satisfies $|N_G[e]\cap L|\ge 2$ and every pair of distinct edges $e_i,e_j$ satisfies $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$. The associated optimization problem, usually called $\textsc{MinLVEDP}$, asks for such a set of minimum cardinality; its minimum value is the liar’s ve-domination number $\gamma_{lve}(G)$ (Bhattacharya et al., 2023). The notion extends classical vertex-edge domination by combining local redundancy with pairwise disambiguation, and subsequent work has developed exact algorithms, hardness results, and approximation guarantees across several graph classes (Bhattacharya et al., 2023, Bhattacharya et al., 7 Jul 2025, Bhattacharya et al., 15 Sep 2025).

1. Formal definition and antecedents

The immediate precursor is ordinary vertex-edge domination. For a graph $G=(V,E)$, a vertex $e=xy$0 ve-dominates an edge $e=xy$1 iff $e=xy$2 or $e=xy$3; equivalently, $e=xy$4 ve-dominates exactly those edges having at least one endpoint in $e=xy$5. A set $e=xy$6 is a ve-dominating set if every edge is ve-dominated by some vertex of $e=xy$7, and the minimum size of such a set is $e=xy$8. The same literature also considers independent ve-dominating sets and the parameter $e=xy$9 (Paul et al., 2019).

A broader multiplicity-based generalization is $N_G[e]=N_G[x]\cup N_G[y]$0-vertex-edge domination. A set $N_G[e]=N_G[x]\cup N_G[y]$1 is a $N_G[e]=N_G[x]\cup N_G[y]$2-vertex-edge dominating set if, for every edge $N_G[e]=N_G[x]\cup N_G[y]$3,

$N_G[e]=N_G[x]\cup N_G[y]$4

The corresponding minimum size is denoted $N_G[e]=N_G[x]\cup N_G[y]$5 (Bhattacharya et al., 2023). In this formulation, liar’s vertex-edge domination can be viewed as a strengthening of the case $N_G[e]=N_G[x]\cup N_G[y]$6, but not as an exact synonym for it.

The liar variant fixes the dominated objects as edges and the dominators as vertices, while adding a pairwise condition. A set $N_G[e]=N_G[x]\cup N_G[y]$7 is a liar’s vertex-edge dominating set if

$N_G[e]=N_G[x]\cup N_G[y]$8

and

$N_G[e]=N_G[x]\cup N_G[y]$9

The optimization problem is the $L\subseteq V$0 ($L\subseteq V$1); the decision version is $L\subseteq V$2 (Bhattacharya et al., 2023).

2. Conceptual meaning and relation to nearby notions

The standard interpretation is in communication networks. Vertices host sentinels and edges represent communication channels. A sentinel at $L\subseteq V$3 can monitor damaged channels incident to vertices in $L\subseteq V$4, so an edge $L\subseteq V$5 is monitored by sentinels in $L\subseteq V$6. Ordinary ve-domination ensures one witness per channel. A $L\subseteq V$7-ve dominating set models tolerance to up to $L\subseteq V$8 sentinel failures near a damaged edge. Liar’s vertex-edge domination addresses a stronger fault model in which a sentinel may correctly detect a broken link but falsely report another nearby undamaged link as broken (Bhattacharya et al., 2023, Bhattacharya et al., 7 Jul 2025).

Under that model, the first liar condition,

$L\subseteq V$9

ensures that every edge has at least two local witnesses. The second condition,

$|N_G[e]\cap L|\ge 2$0

ensures that any two distinct edges together have at least three witnesses, preventing ambiguity generated by a single false report (Bhattacharya et al., 2023).

This places liar’s vertex-edge domination strictly between double and triple vertex-edge domination. The literature states explicitly that every liar’s ve-dominating set is a $|N_G[e]\cap L|\ge 2$1-ve dominating set, and every $|N_G[e]\cap L|\ge 2$2-ve dominating set is a liar’s ve-dominating set (Bhattacharya et al., 2023). A common misconception is therefore to identify liar’s ve-domination with $|N_G[e]\cap L|\ge 2$3-ve domination; the pairwise union condition shows that this is incorrect. A second misconception is to treat it as a verbatim transfer of liar’s domination on vertices. In liar’s ve-domination, the dominated objects are edges, but domination is still realized by vertices, so the relevant neighbourhoods are $|N_G[e]\cap L|\ge 2$4 rather than $|N_G[e]\cap L|\ge 2$5 (Bhattacharya et al., 7 Jul 2025).

3. Exact algorithms on tractable graph classes

The first exact algorithmic result for the liar variant is linear-time solvability on trees. The tree algorithm is presented as a labelling-based greedy reduction algorithm on an $|N_G[e]\cap L|\ge 2$6-labelled tree in which each edge has a label $|N_G[e]\cap L|\ge 2$7 and each vertex has a label $|N_G[e]\cap L|\ge 2$8. A set $|N_G[e]\cap L|\ge 2$9 is an $e_i,e_j$0-dominating set if required red vertices are included, every edge $e_i,e_j$1 satisfies $e_i,e_j$2, and every pair of distinct edges $e_i,e_j$3 satisfies $e_i,e_j$4. The original $e_i,e_j$5 instance is recovered by taking all vertices black and all edge labels equal to $e_i,e_j$6 (Bhattacharya et al., 2023).

The tree is processed bottom-up in a BFS-derived ordering. At each level, support vertices are handled in two rounds. The first round forces enough vertices in a local closed neighbourhood to satisfy edgewise requirements; the second round repairs the pair condition involving the parent edge. Once forced child vertices are committed, the children are deleted and the residual demand on the parent edge is updated by

$e_i,e_j$7

When the residual tree becomes a rooted star, a direct star subroutine completes the solution. The total running time is linear (Bhattacharya et al., 2023).

A later development extends exact linear-time solvability to block graphs. Here the central structure is the cut-tree $e_i,e_j$8, whose nodes represent blocks and cut vertices. The algorithm works on a labelled block graph in which each vertex $e_i,e_j$9 carries a label $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$0, with $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$1 and $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$2 a nonnegative integer, while each edge has a label $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$3. An $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$4-dominating set must include all red vertices, satisfy $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$5 for every vertex, satisfy $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$6 for every edge, and satisfy

$|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$7

for every pair of distinct edges $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$8. Support blocks are processed bottom-up, first resolving internal requirements of descendant end blocks and then resolving conditions attached to the relevant cut vertices. After deleting solved branches, residual labels are updated through formulas such as

$|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$9

$\textsc{MinLVEDP}$0

and the recurrence

$\textsc{MinLVEDP}$1

The final residual case is a star block graph, and the overall running time is $\textsc{MinLVEDP}$2 (Bhattacharya et al., 7 Jul 2025).

Proper interval graphs also admit a linear-time algorithm. The method relies on an interval representation with no interval properly containing another and on an edge interval ordering $\textsc{MinLVEDP}$3. For each current edge $\textsc{MinLVEDP}$4, the algorithm first ensures two witnesses by adding the rightmost needed vertices from $\textsc{MinLVEDP}$5. It then finds the smallest later edge $\textsc{MinLVEDP}$6 that either still has too few selected neighbours or has exactly the same selected neighbourhood contribution as $\textsc{MinLVEDP}$7, and it adds one or two more rightmost vertices accordingly. The correctness proof uses proper interval geometry, especially the fact that a non-neighbouring vertex lies completely to the left or completely to the right of the edge interval, together with exchange lemmas showing that rightmost choices can replace arbitrary witnesses without loss of optimality. The total running time is again $\textsc{MinLVEDP}$8 (Bhattacharya et al., 7 Jul 2025).

4. Hardness and complexity boundaries

Despite these exact algorithms, the decision problem is hard on several natural graph classes. The initial algorithmic study proves that $\textsc{MinLVEDP}$9 is NP-complete for general graphs, chordal graphs, bipartite graphs, and $\gamma_{lve}(G)$0-claw free graphs for $\gamma_{lve}(G)$1 (Bhattacharya et al., 2023). The basic reduction attaches one fresh leaf to each vertex of an input instance of liar’s domination. If $\gamma_{lve}(G)$2 has vertices $\gamma_{lve}(G)$3, then the constructed graph is

$\gamma_{lve}(G)$4

The pendant edge $\gamma_{lve}(G)$5 has $\gamma_{lve}(G)$6, so liar’s domination constraints on vertices are encoded as liar’s ve-domination constraints on pendant edges (Bhattacharya et al., 2023).

A sharper negative result appears for undirected path graphs. The problem remains NP-complete on this class by reduction from 3-Dimensional Matching. The reduction builds a clique tree and shows that the constructed graph has a liar’s ve-dominating set of size

$\gamma_{lve}(G)$7

if and only if the original $\gamma_{lve}(G)$8-DMP instance has a matching of size $\gamma_{lve}(G)$9 (Bhattacharya et al., 7 Jul 2025). This is particularly significant because the same paper also gives linear-time algorithms for block graphs and proper interval graphs, both closely related to chordal structure. The resulting boundary inside chordal-related classes is therefore genuinely nontrivial.

The problem is also NP-complete in unit disk graphs. The reduction starts from planar vertex cover in graphs of maximum degree $G=(V,E)$0, uses an orthogonal grid embedding, transforms each embedded edge into a path of carefully placed unit-distance vertices, and attaches a pendant length-$G=(V,E)$1 segment to every original vertex point. The key threshold is that the unit disk graph $G=(V,E)$2 has a liar’s ve-dominating set of size at most

$G=(V,E)$3

if and only if the original planar graph has a vertex cover of size at most $G=(V,E)$4, where $G=(V,E)$5 is the number of original vertices and $G=(V,E)$6 is the total number of line segments in the orthogonal embedding (Bhattacharya et al., 15 Sep 2025).

5. Approximation algorithms, lower bounds, and geometric approximation

For general graphs, the principal approximation algorithm is two-phase. First, it computes an approximate $G=(V,E)$7-ve dominating set $G=(V,E)$8 satisfying

$G=(V,E)$9

where $e=xy$00 is an optimum $e=xy$01-ve dominating set. If $e=xy$02 already satisfies the liar pair condition, it is accepted. Otherwise, the algorithm forms the universe

$e=xy$03

of deficient edge pairs and, for each vertex $e=xy$04, the set

$e=xy$05

A greedy set cover on $e=xy$06 produces a repair set $e=xy$07, and $e=xy$08 becomes a liar’s ve-dominating set. The final guarantee is

$e=xy$09

so $e=xy$10 admits a polynomial-time $e=xy$11-approximation (Bhattacharya et al., 2023).

For $e=xy$12-claw free graphs, the approximation method is combinatorial rather than set-cover-based. It computes three maximal distance-$e=xy$13 matchings $e=xy$14 in succession and returns

$e=xy$15

The analysis uses the lower bound

$e=xy$16

for a maximal distance-$e=xy$17 matching $e=xy$18 and a minimum $e=xy$19-ve dominating set $e=xy$20. In the liar case this yields

$e=xy$21

from which the algorithm obtains a $e=xy$22-approximation (Bhattacharya et al., 2023).

These positive results are paired with hardness of approximation. The problem cannot be approximated within

$e=xy$23

for any $e=xy$24, unless

$e=xy$25

It is also APX-complete for graphs with maximum degree $e=xy$26 and for $e=xy$27-claw free graphs for $e=xy$28 (Bhattacharya et al., 2023).

Unit disk graphs behave differently. Although exact solution is NP-complete, the minimum problem admits a PTAS. The method decomposes the graph into $e=xy$29-separated local regions, expands each region to a bounded-radius neighbourhood, computes local optima exactly, and glues them together. A useful auxiliary fact is that the union of three successive maximal independent sets is always a liar’s ve-dominating set. For a radius-$e=xy$30 neighbourhood, this yields the explicit bound

$e=xy$31

The algorithm chooses the smallest radius $e=xy$32 such that

$e=xy$33

and the stopping radius satisfies

$e=xy$34

Because these local neighbourhoods have bounded size for fixed $e=xy$35, local optima can be found by brute force, and their union yields a $e=xy$36-approximation with running time

$e=xy$37

(Bhattacharya et al., 15 Sep 2025).

6. Structural themes, methodology, and current research frontiers

Several methodological patterns recur across the literature. One is labelled reduction: the tree algorithm uses edge demands $e=xy$38 and vertex colours $e=xy$39, while the block-graph algorithm uses the richer state space $e=xy$40 together with edge labels $e=xy$41. In both cases, partial solutions are summarized as residual local obligations rather than recomputed globally (Bhattacharya et al., 2023, Bhattacharya et al., 7 Jul 2025).

A second pattern is the use of graph-specific decompositions. Trees are handled bottom-up through support vertices; block graphs through cut-trees, end blocks, and support blocks; proper interval graphs through an edge interval ordering; unit disk graphs through separated neighbourhoods and packing bounds (Bhattacharya et al., 2023, Bhattacharya et al., 7 Jul 2025, Bhattacharya et al., 15 Sep 2025). This indicates that the tractable cases identified so far depend heavily on highly constrained structure.

A third theme is the interaction between local multiplicity and pairwise separation. The $e=xy$42-ve framework captures the redundancy component alone, whereas liar’s ve-domination adds an explicit condition on pairs of edges (Bhattacharya et al., 2023, Bhattacharya et al., 2023). The approximation algorithm for general graphs makes this split especially transparent: a first phase enforces local double coverage, and a second phase repairs deficient edge pairs through a set-cover instance (Bhattacharya et al., 2023).

Background work on ordinary ve-domination also supplies a cautionary perspective. A claimed reduction from ve-domination on trees to weighted distance-$e=xy$43 domination was shown to fail, and ordinary ve-domination already exhibits delicate behaviour on trees, block graphs, and undirected path graphs (Paul et al., 2019). This suggests that liar’s ve-domination, which depends not only on coverage but also on coverage multiplicity across pairs of edges, is structurally stricter than a naive radius-based reformulation would capture.

The current frontier is therefore not the existence of a single universal technique, but the delineation of graph classes where the liar constraints remain compressible. The recent literature explicitly points toward further study on other subclasses of chordal graphs, on bipartite and other graph classes, and on additional approximation algorithms beyond the presently known settings (Bhattacharya et al., 7 Jul 2025, Bhattacharya et al., 15 Sep 2025).