LMRTTG: Locally Most Reliable Two-Terminal Graph

• The paper shows that the LMRTTG uniquely maximizes two-terminal reliability for small p by lexicographically optimizing the spanning subgraph count coefficients.
• It categorizes graph constructions into sparse and dense regimes, using bipartite and universal terminal designs to achieve optimal connectivity.
• The findings imply practical applications in network optimization and combinatorial design, setting benchmarks for reliability analysis in resilient systems.

A Locally Most Reliable Two-Terminal Graph (LMRTTG) is a two-terminal graph structure that, for fixed numbers of vertices $n$ and edges $m$, uniquely maximizes the probability of connectivity between designated terminals $s$ and $t$ in the regime of small edge activation probability $p$. This optimality is "local" in the sense that the graph exhibits the highest two-terminal reliability within a neighborhood of $p = 0$ compared to all other graphs in the set $T_{n,m}$ of nonisomorphic two-terminal graphs with fixed $n$ and $m$ (Romero, 20 Sep 2025). LMRTTGs have significance in canonical reliability models, network optimization, and combinatorial design, and their existence and structural characteristics have been fully characterized for all $n \geq 4$ and $5 \leq m \leq \binom{n}{2}$. Below, the fundamental definitions, theoretical results, and construction principles are detailed.

1. Two-Terminal Reliability and the LMRTTG Criterion

For a two-terminal graph $G$ on $n$ vertices and $m$ edges with terminals $s$ and $t$, the two-terminal reliability function is

$R_G(p) = \sum_{i=1}^m N_i(G) \, p^i (1-p)^{m-i}$

where $N_i(G)$ is the number of spanning subgraphs with exactly $i$ edges in which $s$ and $t$ are connected. The Locally Most Reliable Two-Terminal Graph property is defined as follows: $G$ is an LMRTTG in $T_{n,m}$ if, for every $H \in T_{n,m}$, there exists $\delta > 0$ such that for all $p \in (0, \delta)$,

$R_G(p) \geq R_H(p)$

The underlying combinatorial principle is lexicographic optimization: $G$'s coefficient vector $(N_1(G), N_2(G), \ldots, N_m(G))$ is lex maximal in $T_{n,m}$ ($$N_1(G) = N_1(H), \ldots, N_{j-1}(G) = N_{j-1}(H), N_j(G) > N_j(H)$$ implies $G$ is better in a neighborhood of $p=0$).

2. Existence and Uniqueness in $(n, m)$-Classes

The existence and uniqueness of LMRTTGs have been established for all $n \geq 4$ and $5 \leq m \leq \binom{n}{2}$ (Romero, 20 Sep 2025):

• For $5 \leq m \leq 2n-3$: There is a unique LMRTTG, explicitly constructed as a bipartite graph ("quasi-star") where each terminal is adjacent to $(m-1)/2$ vertices if $m$ is odd, or $(m-2)/2$ vertices and an extra edge $v_1v_2$ if $m$ is even.
• For $2n-3 < m \leq \binom{n}{2}$: Both terminals are universal (adjacent to all other vertices), and the remaining subgraph $\hat{G}$ on $n-2$ vertices and $m-(2n-3)$ edges must optimize a graph invariant

$H(\hat{G}) = M_2(\hat{G}) - 6k_3(\hat{G})$

where $M_2$ is the second Zagreb index and $k_3$ is the number of triangles. The LMRTTG is constructed by taking the unique $H$-optimal graph $\hat{G}$ and joining both terminals universally.

This dichotomy is proven by recursive application of lexicographic maximization and combinatorial invariants for coefficient vectors.

3. Analytical and Combinatorial Construction Methods

Sparse Regime ($5 \leq m \leq 2n-3$)

Set $V = \{s, t, v_1, ..., v_{n-2}\}$. Then:

• $m$ odd: $$E(G_{n,m}) = \{st\} \cup \{sv_i, v_it: 1 \leq i \leq (m-1)/2\}$$
• $m$ even: $$E(G_{n,m}) = \{st\} \cup \{sv_i, v_it: 1 \leq i \leq (m-2)/2\} \cup \{v_1v_2\}$$

Dense Regime ($2n-3 < m \leq \binom{n}{2}$)

Construct $G_{n,m}$ such that both $s$ and $t$ are universal, and

$G_{n,m} - \{s, t\} = H_{n-2, m-(2n-3)}$

where $H_{n-2, m-(2n-3)}$ is the unique $H$-optimal simple graph for these parameters.

The definition of $H$-optimality is fully combinatorial and uses polynomial expressions and bounds on the number of triangles and degree-squared sums (Zagreb index).

4. Comparison with Uniform and Split Reliability Notions

LMRTTG describes a local optimality at $p \to 0$ for connectivity between two terminals. By contrast, uniformly most reliable graphs would require optimality for all $p\in[0,1]$. Uniform optimality often fails except in special cases (e.g., the complete graph minus a non-terminal edge for very dense graphs (Xie et al., 2019)).

A similar principle appears in the paper of locally and uniformly most split reliable two-terminal graphs (LMRTTG analogues for split reliability), where optimal graphs are described via split equivalence to the balloon graph and can exist only in enumerated regimes (Romero, 19 Mar 2025).

5. Role of Parameters $n$ and $m$

The structure of LMRTTG changes categorically as $m$ increases for fixed $n$:

• Sparse constructions: Terminal-to-vertex bipartite skeleton for $m$ just above minimal connectivity.
• Dense constructions: Universal terminals with highly regular or $H$-optimal inner subgraphs; the combinatorial design becomes sensitive to the target invariant $H$.

This parameter-driven classification encodes the fundamental trade-off between sparsity, redundancy, and triangle suppression (via $H$), controlling reliability expansion and local optimality.

6. Algebraic and Polynomial Properties

LMRTTG selection is fundamentally rooted in maximizing the reliability polynomial's coefficients for small $p$: $\text{If %%%%70%%%% is lex maximal, } \exists \delta \ \text{such that } R_G(p) \geq R_H(p) \ \forall p \in (0,\delta)$ This algebraic approach is canonical and extends to analyzing polynomial root locations, phase transitions (as in ladder networks 0701005), and the sensitivity of reliability near $p=0$ (where only the lowest-degree terms matter).

7. Broader Implications and Research Directions

Identification and construction of LMRTTGs provide robust benchmarks for network reliability analysis, inform combinatorial optimization algorithms, and suggest new ways to design resilient communication architectures where local connectivity is paramount. The lexicographic coefficient maximization method and $H$-optimality framework may generalize to multi-terminal, split, and $d$-constrained reliability problems (Romero, 28 Apr 2025). Open research continues into the uniqueness and combinatorial properties of LMRTTG analogues for other reliability models, the impact of global graph invariants, and connections to optimal network synthesis principles in the presence of probabilistic failures.

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In summary, the LMRTTG is now a precisely characterized graph structure for all feasible pairs $(n, m)$, established via lexicographic reliability polynomial maximization and combinatorial invariants, with explicit construction principles and implications for network reliability optimization and design (Romero, 20 Sep 2025).